{"id":8986,"date":"2017-04-27T22:04:59","date_gmt":"2017-04-27T22:04:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&p=8986"},"modified":"2024-05-02T16:59:22","modified_gmt":"2024-05-02T16:59:22","slug":"problem-set-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/problem-set-real-numbers\/","title":{"raw":"Problem Set: Real Numbers","rendered":"Problem Set: Real Numbers"},"content":{"raw":"

Classes of Real Numbers<\/h3>\r\n

Rational Numbers<\/strong><\/h4>\r\nIn the following exercises, write as the ratio of two integers.\r\n
    \r\n \t
  1. \u24d0 [latex]5[\/latex]<\/li>\r\n \t
  2. \u24d1 [latex]3.19[\/latex]<\/li>\r\n<\/ol>\r\n
      \r\n \t
    1. \u24d0 [latex]\\Large\\frac{5}{1}[\/latex]<\/li>\r\n \t
    2. \u24d1 [latex]\\Large\\frac{319}{100}[\/latex]<\/li>\r\n<\/ol>\r\n
        \r\n \t
      1. \u24d0 [latex]8[\/latex]<\/li>\r\n \t
      2. \u24d1 [latex]-1.61[\/latex]<\/li>\r\n<\/ol>\r\n
          \r\n \t
        1. \u24d0 [latex]-12[\/latex]<\/li>\r\n \t
        2. \u24d1 [latex]9.279[\/latex]<\/li>\r\n<\/ol>\r\n
            \r\n \t
          1. \u24d0 [latex]\\Large\\frac{-12}{1}[\/latex]<\/li>\r\n \t
          2. \u24d1 [latex]\\Large\\frac{9279}{1000}[\/latex]<\/li>\r\n<\/ol>\r\n
              \r\n \t
            1. \u24d0 [latex]-16[\/latex]<\/li>\r\n \t
            2. \u24d1 [latex]4.399[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercises, determine which of the given numbers are rational and which are irrational.\r\n\r\n[latex]0.75[\/latex] , [latex]0.22\\stackrel{\\text{-}}{\\text{3}}[\/latex] , [latex]\\text{1.39174... }[\/latex]\r\n\r\nRational: [latex]0.75,0.22\\stackrel{\\text{-}}{\\text{3}}[\/latex] . Irrational: [latex]\\text{1.39174... }[\/latex]\r\n\r\n[latex]0.36[\/latex] , [latex]\\text{0.94729\\ldots }[\/latex] , [latex]2.52\\stackrel{\\text{-}}{\\text{8}}[\/latex]\r\n\r\n[latex]0.\\stackrel{\\text{-}}{45}[\/latex] , [latex]\\text{1.919293... }[\/latex] , [latex]3.59[\/latex]\r\n\r\nRational: [latex]0.\\stackrel{\\text{-}}{45}[\/latex] , [latex]3.59[\/latex] . Irrational: [latex]\\text{1.919293... }[\/latex]\r\n\r\n[latex]0.1\\stackrel{\\text{-}}{\\text{3}},\\text{0.42982... }[\/latex] , [latex]1.875[\/latex]\r\n\r\nIn the following exercises, identify whether each number is rational or irrational.\r\n
                \r\n \t
              1. \u24d0 [latex]\\sqrt{25}[\/latex]<\/li>\r\n \t
              2. \u24d1 [latex]\\sqrt{30}[\/latex]<\/li>\r\n<\/ol>\r\n
                  \r\n \t
                1. \u24d0 rational<\/li>\r\n \t
                2. \u24d1 irrational<\/li>\r\n<\/ol>\r\n
                    \r\n \t
                  1. \u24d0 [latex]\\sqrt{44}[\/latex]<\/li>\r\n \t
                  2. \u24d1 [latex]\\sqrt{49}[\/latex]<\/li>\r\n<\/ol>\r\n
                      \r\n \t
                    1. \u24d0 [latex]\\sqrt{164}[\/latex]<\/li>\r\n \t
                    2. \u24d1 [latex]\\sqrt{169}[\/latex]<\/li>\r\n<\/ol>\r\n
                        \r\n \t
                      1. \u24d0 irrational<\/li>\r\n \t
                      2. \u24d1 rational<\/li>\r\n<\/ol>\r\n
                          \r\n \t
                        1. \u24d0 [latex]\\sqrt{225}[\/latex]<\/li>\r\n \t
                        2. \u24d1 [latex]\\sqrt{216}[\/latex]<\/li>\r\n<\/ol>\r\n

                          Classifying Real Numbers<\/strong><\/h3>\r\nIn the following exercises, determine whether each number is whole, integer, rational, irrational, and real.\r\n\r\n[latex]-8[\/latex] , [latex]0,\\text{1.95286....}[\/latex] , [latex]\\Large\\frac{12}{5}[\/latex] , [latex]\\sqrt{36}[\/latex] , [latex]9[\/latex]\r\n\r\n\"The\r\n\r\n[latex]-9[\/latex] , [latex]-3\\Large\\frac{4}{9}[\/latex] , [latex]-\\sqrt{9}[\/latex] , [latex]0.4\\stackrel{\\text{-}}{09}[\/latex] , [latex]\\Large\\frac{11}{6}[\/latex] , [latex]7[\/latex]\r\n\r\n[latex]-\\sqrt{100}[\/latex] , [latex]-7[\/latex] , [latex]-\\Large\\frac{8}{3}[\/latex] , [latex]-1[\/latex] , [latex]0.77[\/latex] , [latex]3\\Large\\frac{1}{4}[\/latex]\r\n\r\n\"The\r\n
                          \r\n

                          EVERYDAY MATH<\/h3>\r\n

                          Field trip<\/strong><\/h4>\r\nAll the [latex]5\\text{th}[\/latex] graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be [latex]147[\/latex] people. Each bus holds [latex]44[\/latex] people.\r\n\u24d0 How many buses will be needed?\r\n\u24d1 Why must the answer be a whole number?\r\n\u24d2 Why shouldn't you round the answer the usual way?\r\n

                          Child care<\/strong><\/h4>\r\nSerena wants to open a licensed child care center. Her state requires that there be no more than [latex]12[\/latex] children for each teacher. She would like her child care center to serve [latex]40[\/latex] children.\r\n\u24d0 How many teachers will be needed?\r\n\u24d1 Why must the answer be a whole number?\r\n\u24d2 Why shouldn't you round the answer the usual way?\r\n\r\n\u24d0 4\r\n\u24d1 Teachers cannot be divided\r\n\u24d2 It would result in a lower number.\r\n

                          <\/h2>\r\n<\/div>\r\n \r\n
                          \r\n

                          WRITING EXERCISES<\/h3>\r\nIn your own words, explain the difference between a rational number and an irrational number.\r\n\r\nExplain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.\r\n\r\nAnswers will vary.\r\n\r\n<\/div>\r\n \r\n

                          <\/h2>\r\n

                          Using the Commutative and Associative Properties<\/h3>\r\nIn the following exercises, use the commutative properties to rewrite the given expression.\r\n\r\n[latex]\\text{8+9=___}[\/latex]\r\n\r\n[latex]\\text{7+6=___}[\/latex]\r\n\r\n7 + 6 = 6 + 7\r\n\r\n[latex]\\text{8}\\left(-12\\right)\\text{=___}[\/latex]\r\n\r\n[latex]\\text{7}\\left(-13\\right)\\text{=___}[\/latex]\r\n\r\n7(\u221213) = (\u221213)7\r\n\r\n[latex]\\text{(-19)(-14)=___}[\/latex]\r\n\r\n[latex]\\left(-12\\right)\\left(-18\\right)\\text{=___}[\/latex]\r\n\r\n(\u221212)(\u221218) = (\u221218)(\u221212)\r\n\r\n[latex]\\text{-11+8=___}[\/latex]\r\n\r\n[latex]\\text{-15+7=___}[\/latex]\r\n\r\n\u221215 + 7 = 7 + (\u221215)\r\n\r\n[latex]\\text{x+4=___}[\/latex]\r\n\r\n[latex]\\text{y+1=___}[\/latex]\r\n\r\ny<\/em> + 1 = 1 + y<\/em>\r\n\r\n[latex]\\text{-2a=___}[\/latex]\r\n\r\n[latex]\\text{-3m=___}[\/latex]\r\n\r\n\u22123m<\/em> = m<\/em>(\u22123)\r\n\r\nIn the following exercises, use the associative properties to rewrite the given expression.\r\n\r\n[latex]\\left(11+9\\right)\\text{+14=___}[\/latex]\r\n\r\n[latex]\\left(21+14\\right)\\text{+9=___}[\/latex]\r\n\r\n(21 + 14) + 9 = 21 + (14 + 9)\r\n\r\n[latex]\\left(12\\cdot 5\\right)\\cdot \\text{7=___}[\/latex]\r\n\r\n[latex]\\left(14\\cdot 6\\right)\\cdot \\text{9=___}[\/latex]\r\n\r\n(14 \u00b7 6) \u00b7 9 = 14(6 \u00b7 9)\r\n\r\n[latex]\\left(-7+9\\right)\\text{+8=___}[\/latex]\r\n\r\n[latex]\\left(-2+6\\right)\\text{+7=___}[\/latex]\r\n\r\n(\u22122 + 6) + 7 = \u22122 + (6 + 7)\r\n\r\n[latex]\\left(16\\cdot\\Large\\frac{4}{5}\\normalsize\\right)\\cdot \\text{15=___}[\/latex]\r\n\r\n[latex]\\left(13\\cdot\\Large\\frac{2}{3}\\normalsize\\right)\\cdot \\text{18=___}[\/latex]\r\n\r\n[latex]\\left(13\\cdot\\Large\\frac{2}{3}\\normalsize\\right)\\cdot 18=13\\left(\\Large\\frac{2}{3}\\normalsize\\cdot 18\\right)[\/latex]\r\n\r\n[latex]3\\left(4x\\right)\\text{=___}[\/latex]\r\n\r\n[latex]4\\left(7x\\right)\\text{=___}[\/latex]\r\n\r\n4(7x<\/em>) = (4 \u00b7 7)x<\/em>\r\n\r\n[latex]\\left(12+x\\right)\\text{+28=___}[\/latex]\r\n\r\n[latex]\\left(17+y\\right)\\text{+33=___}[\/latex]\r\n\r\n(17 + y<\/em>) + 33 = 17 + (y<\/em> + 33)\r\n

                          Evaluate Expressions using the Commutative and Associative Properties<\/strong><\/h3>\r\nIn the following exercises, evaluate each expression for the given value.\r\n\r\nIf [latex]y=\\Large\\frac{5}{8}[\/latex], evaluate:\r\n\r\n\u24d0 [latex]y+0.49+\\left(-y\\right)[\/latex]\r\n\u24d1 [latex]y+\\left(-y\\right)+0.49[\/latex]\r\n\r\nIf [latex]z=\\Large\\frac{7}{8}[\/latex], evaluate:\r\n\r\n\u24d0 [latex]z+0.97+\\left(-z\\right)[\/latex]\r\n\u24d1 [latex]z+\\left(-z\\right)+0.97[\/latex]\r\n
                            \r\n \t
                          1. \u24d0 0.97<\/li>\r\n \t
                          2. \u24d1 0.97<\/li>\r\n<\/ol>\r\nIf [latex]c=-\\Large\\frac{11}{4}[\/latex], evaluate:\r\n\r\n\u24d0 [latex]c+3.125+\\left(-c\\right)[\/latex]\r\n\u24d1 [latex]c+\\left(-c\\right)+3.125[\/latex]\r\n\r\nIf [latex]d=-\\Large\\frac{9}{4}[\/latex], evaluate:\r\n\r\n\u24d0 [latex]d+2.375+\\left(-d\\right)[\/latex]\r\n\u24d1 [latex]d+\\left(-d\\right)+2.375[\/latex]\r\n\r\n\u24d0 2.375\r\n\u24d1 2.375\r\n\r\nIf [latex]j=11[\/latex], evaluate:\r\n\r\n\u24d0 [latex]\\Large\\frac{5}{6}\\left(\\frac{6}{5}\\normalsize j\\Large\\right)[\/latex]\r\n\u24d1 [latex]\\Large\\left(\\frac{5}{6}\\normalsize\\cdot\\Large\\frac{6}{5}\\right)\\normalsize j[\/latex]\r\n\r\nIf [latex]k=21[\/latex], evaluate:\r\n\r\n\u24d0 [latex]\\Large\\frac{4}{13}\\left(\\frac{13}{4}\\normalsize k\\Large\\right)[\/latex]\r\n\u24d1 [latex]\\Large\\left(\\frac{4}{13}\\normalsize\\cdot\\Large\\frac{13}{4}\\right)\\normalsize k[\/latex]\r\n\r\n\u24d0 21\r\n\u24d1 21\r\n\r\nIf [latex]m=-25[\/latex], evaluate:\r\n\r\n\u24d0 [latex]-\\Large\\frac{3}{7}\\left(\\frac{7}{3}\\normalsize m\\Large\\right)[\/latex]\r\n\u24d1 [latex]\\Large\\left(-\\frac{3}{7}\\normalsize\\cdot\\Large\\frac{7}{3}\\right)\\normalsize m[\/latex]\r\n\r\nIf [latex]n=-8[\/latex], evaluate:\r\n\r\n\u24d0 [latex]-\\Large\\frac{5}{21}\\left(\\frac{21}{5}\\normalsize n\\Large\\right)[\/latex]\r\n\u24d1 [latex]\\Large\\left(-\\frac{5}{21}\\normalsize\\cdot\\Large\\frac{21}{5}\\right)\\normalsize n[\/latex]\r\n\r\n\u24d0 \u22128\r\n\u24d1 \u22128\r\n

                            Simplify Expressions Using the Commutative and Associative Properties<\/strong><\/h3>\r\nIn the following exercises, simplify.\r\n\r\n[latex]-45a+15+45a[\/latex]\r\n\r\n[latex]9y+23+\\left(-9y\\right)[\/latex]\r\n\r\n23\r\n\r\n[latex]\\Large\\frac{1}{2}\\normalsize +\\Large\\frac{7}{8}\\normalsize +\\Large\\left(-\\frac{1}{2}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{2}{5}\\normalsize +\\Large\\frac{5}{12}\\normalsize+\\Large\\left(-\\frac{2}{5}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{12}[\/latex]\r\n\r\n[latex]\\Large\\frac{3}{20}\\normalsize\\cdot\\Large\\frac{49}{11}\\normalsize\\cdot\\Large\\frac{20}{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{13}{18}\\normalsize\\cdot\\Large\\frac{25}{7}\\normalsize\\cdot\\Large\\frac{18}{13}[\/latex]\r\n\r\n[latex]\\Large\\frac{25}{7}[\/latex]\r\n\r\n[latex]\\Large\\frac{7}{12}\\normalsize\\cdot\\Large\\frac{9}{17}\\normalsize\\cdot\\Large\\frac{24}{7}[\/latex]\r\n\r\n[latex]\\Large\\frac{3}{10}\\normalsize\\cdot\\Large\\frac{13}{23}\\normalsize\\cdot\\Large\\frac{50}{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{65}{23}[\/latex]\r\n\r\n[latex]-24\\cdot 7\\cdot\\Large\\frac{3}{8}[\/latex]\r\n\r\n[latex]-36\\cdot 11\\cdot\\Large\\frac{4}{9}[\/latex]\r\n\r\n\u2212176\r\n\r\n[latex]\\Large\\left(\\frac{5}{6}\\normalsize +\\Large\\frac{8}{15}\\right)\\normalsize +\\Large\\frac{7}{15}[\/latex]\r\n\r\n[latex]\\Large\\left(\\frac{1}{12}\\normalsize +\\Large\\frac{4}{9}\\right)\\normalsize +\\Large\\frac{5}{9}[\/latex]\r\n\r\n[latex]\\Large\\frac{13}{12}[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{13}\\normalsize +\\Large\\frac{3}{4}\\normalsize +\\Large\\frac{1}{4}[\/latex]\r\n\r\n[latex]\\Large\\frac{8}{15}\\normalsize +\\Large\\frac{5}{7}\\normalsize +\\Large\\frac{2}{7}[\/latex]\r\n\r\n[latex]\\Large\\frac{23}{15}[\/latex]\r\n\r\n[latex]\\left(4.33p+1.09p\\right)+3.91p[\/latex]\r\n\r\n[latex]\\left(5.89d+2.75d\\right)+1.25d[\/latex]\r\n\r\n9.89d<\/em>\r\n\r\n[latex]17\\left(0.25\\right)\\left(4\\right)[\/latex]\r\n\r\n[latex]36\\left(0.2\\right)\\left(5\\right)[\/latex]\r\n\r\n36\r\n\r\n[latex]\\left[2.48\\left(12\\right)\\right]\\left(0.5\\right)[\/latex]\r\n\r\n[latex]\\left[9.731\\left(4\\right)\\right]\\left(0.75\\right)[\/latex]\r\n\r\n29.193\r\n\r\n[latex]7\\left(4a\\right)[\/latex]\r\n\r\n[latex]9\\left(8w\\right)[\/latex]\r\n\r\n72w<\/em>\r\n\r\n[latex]-15\\left(5m\\right)[\/latex]\r\n\r\n[latex]-23\\left(2n\\right)[\/latex]\r\n\r\n\u221246n<\/em>\r\n\r\n[latex]12\\left(\\Large\\frac{5}{6}\\normalsize p\\right)[\/latex]\r\n\r\n[latex]20\\left(\\Large\\frac{3}{5}\\normalsize q\\right)[\/latex]\r\n\r\n12q<\/em>\r\n\r\n[latex]14x+19y+25x+3y[\/latex]\r\n\r\n[latex]15u+11v+27u+19v[\/latex]\r\n\r\n42u<\/em> + 30v<\/em>\r\n\r\n[latex]43m+\\left(-12n\\right)+\\text{ }\\left(-16m\\right)+\\left(-9n\\right)[\/latex]\r\n\r\n[latex]-22p+17q+\\left(-35p\\right)+\\left(-27q\\right)[\/latex]\r\n\r\n\u221257p<\/em> + (\u221210q<\/em>)\r\n\r\n[latex]\\Large\\frac{3}{8}\\normalsize g+\\Large\\frac{1}{12}\\normalsize h+\\Large\\frac{7}{8}\\normalsize g+\\Large\\frac{5}{12}\\normalsize h[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{6}\\normalsize a+\\Large\\frac{3}{10}\\normalsize b+\\Large\\frac{1}{6}\\normalsize a+\\Large\\frac{9}{10}\\normalsize b[\/latex]\r\n\r\n[latex]a+\\Large\\frac{6}{5}\\normalsize b[\/latex]\r\n\r\n[latex]6.8p+9.14q+\\left(-4.37p\\right)+\\left(-0.88q\\right)[\/latex]\r\n\r\n[latex]9.6m+7.22n+\\left(-2.19m\\right)+\\left(-0.65n\\right)[\/latex]\r\n\r\n7.41m<\/em> + 6.57n<\/em>\r\n
                            \r\n

                            EVERYDAY MATH<\/h3>\r\n

                            Stamps<\/strong><\/h4>\r\nAllie and Loren need to buy stamps. Allie needs four [latex]{$0.49}[\/latex] stamps and nine [latex]{$0.02}[\/latex] stamps. Loren needs eight [latex]{$0.49}[\/latex] stamps and three [latex]{$0.02}[\/latex] stamps.\r\n\u24d0 How much will Allie\u2019s stamps cost?\r\n\u24d1 How much will Loren\u2019s stamps cost?\r\n\u24d2 What is the total cost of the girls\u2019 stamps?\r\n\u24d3 How many [latex]{$0.49}[\/latex] stamps do the girls need altogether? How much will they cost?\r\n\u24d4 How many [latex]{$0.02}[\/latex] stamps do the girls need altogether? How much will they cost?\r\n

                            Counting Cash<\/strong><\/h4>\r\nGrant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three [latex]{$5}[\/latex] bills, eighteen [latex]{$10}[\/latex] bills, and thirty-four [latex]{$20}[\/latex] bills. In another envelope, he has fourteen [latex]{$5}[\/latex] bills, nine [latex]{$10}[\/latex] bills, and twenty-seven [latex]{$20}[\/latex] bills.\r\n\u24d0 How much money is in the first envelope?\r\n\u24d1 How much money is in the second envelope?\r\n\u24d2 What is the total value of all the cash?\r\n\u24d3 What is the value of all the [latex]{$5}[\/latex] bills?\r\n\u24d4 What is the value of all [latex]{$10}[\/latex] bills?\r\n\u24d5 What is the value of all [latex]{$20}[\/latex] bills?\r\n
                              \r\n \t
                            1. \u24d0 $975<\/li>\r\n \t
                            2. \u24d1 $700<\/li>\r\n \t
                            3. \u24d2 $1675<\/li>\r\n \t
                            4. \u24d3 $185<\/li>\r\n \t
                            5. \u24d4 $270<\/li>\r\n \t
                            6. \u24d5 $1220<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                              \r\n

                              WRITING EXERCISES<\/h3>\r\nIn your own words, state the Commutative Property of Addition and explain why it is useful.\r\n\r\nIn your own words, state the Associative Property of Multiplication and explain why it is useful.\r\n\r\nAnswers will vary.\r\n

                              <\/h2>\r\n<\/div>\r\n \r\n

                              Using the Distributive Property<\/h2>\r\n

                              Simplify Expressions Using the Distributive Property<\/strong><\/h3>\r\nIn the following exercises, simplify using the distributive property.\r\n\r\n[latex]4\\left(x+8\\right)[\/latex]\r\n\r\n[latex]3\\left(a+9\\right)[\/latex]\r\n\r\n3a<\/em> + 27\r\n\r\n[latex]8\\left(4y+9\\right)[\/latex]\r\n\r\n[latex]9\\left(3w+7\\right)[\/latex]\r\n\r\n27w<\/em> + 63\r\n\r\n[latex]6\\left(c - 13\\right)[\/latex]\r\n\r\n[latex]7\\left(y - 13\\right)[\/latex]\r\n\r\n7y<\/em> \u2212 91\r\n\r\n[latex]7\\left(3p - 8\\right)[\/latex]\r\n\r\n[latex]5\\left(7u - 4\\right)[\/latex]\r\n\r\n35u<\/em> \u2212 20\r\n\r\n[latex]\\Large\\frac{1}{2}\\normalsize\\left(n+8\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{3}\\normalsize\\left(u+9\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{3}\\normalsize u+3[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{4}\\normalsize\\left(3q+12\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{5}\\normalsize\\left(4m+20\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{4}{5}\\normalsize m+4[\/latex]\r\n\r\n[latex]9\\left(\\Large\\frac{5}{9}\\normalsize y-\\Large\\frac{1}{3}\\normalsize\\right)[\/latex]\r\n\r\n[latex]10\\left(\\Large\\frac{3}{10}\\normalsize x-\\Large\\frac{2}{5}\\normalsize\\right)[\/latex]\r\n\r\n3x<\/em> \u2212 4\r\n\r\n[latex]12\\left(\\Large\\frac{1}{4}\\normalsize+\\Large\\frac{2}{3}\\normalsize r\\right)[\/latex]\r\n\r\n[latex]12\\left(\\Large\\frac{1}{6}\\normalsize +\\Large\\frac{3}{4}\\normalsize s\\right)[\/latex]\r\n\r\n2 + 9s<\/em>\r\n\r\n[latex]r\\left(s - 18\\right)[\/latex]\r\n\r\n[latex]u\\left(v - 10\\right)[\/latex]\r\n\r\nuv<\/em> \u2212 10u<\/em>\r\n\r\n[latex]\\left(y+4\\right)p[\/latex]\r\n\r\n[latex]\\left(a+7\\right)x[\/latex]\r\n\r\nax<\/em> + 7x<\/em>\r\n\r\n[latex]-2\\left(y+13\\right)[\/latex]\r\n\r\n[latex]-3\\left(a+11\\right)[\/latex]\r\n\r\n\u22123a<\/em> \u2212 33\r\n\r\n[latex]-7\\left(4p+1\\right)[\/latex]\r\n\r\n[latex]-9\\left(9a+4\\right)[\/latex]\r\n\r\n\u221281a<\/em> \u2212 36\r\n\r\n[latex]-3\\left(x - 6\\right)[\/latex]\r\n\r\n[latex]-4\\left(q - 7\\right)[\/latex]\r\n\r\n\u22124q<\/em> + 28\r\n\r\n[latex]-9\\left(3a - 7\\right)[\/latex]\r\n\r\n[latex]-6\\left(7x - 8\\right)[\/latex]\r\n\r\n\u221242x<\/em> + 48\r\n\r\n[latex]-\\left(r+7\\right)[\/latex]\r\n\r\n[latex]-\\left(q+11\\right)[\/latex]\r\n\r\n\u2212q<\/em> \u2212 11\r\n\r\n[latex]-\\left(3x - 7\\right)[\/latex]\r\n\r\n[latex]-\\left(5p - 4\\right)[\/latex]\r\n\r\n\u22125p<\/em> + 4\r\n\r\n[latex]5+9\\left(n - 6\\right)[\/latex]\r\n\r\n[latex]12+8\\left(u - 1\\right)[\/latex]\r\n\r\n8u<\/em> + 4\r\n\r\n[latex]16 - 3\\left(y+8\\right)[\/latex]\r\n\r\n[latex]18 - 4\\left(x+2\\right)[\/latex]\r\n\r\n\u22124x<\/em> + 10\r\n\r\n[latex]4 - 11\\left(3c - 2\\right)[\/latex]\r\n\r\n[latex]9 - 6\\left(7n - 5\\right)[\/latex]\r\n\r\n\u221242n<\/em> + 39\r\n\r\n[latex]22-\\left(a+3\\right)[\/latex]\r\n\r\n[latex]8-\\left(r - 7\\right)[\/latex]\r\n\r\n\u2212r<\/em> + 15\r\n\r\n[latex]-12-\\left(u+10\\right)[\/latex]\r\n\r\n[latex]-4-\\left(c - 10\\right)[\/latex]\r\n\r\n\u2212c<\/em> + 6\r\n\r\n[latex]\\left(5m - 3\\right)-\\left(m+7\\right)[\/latex]\r\n\r\n[latex]\\left(4y - 1\\right)-\\left(y - 2\\right)[\/latex]\r\n\r\n3y<\/em> + 1\r\n\r\n[latex]5\\left(2n+9\\right)+12\\left(n - 3\\right)[\/latex]\r\n\r\n[latex]9\\left(5u+8\\right)+2\\left(u - 6\\right)[\/latex]\r\n\r\n47u<\/em> + 60\r\n\r\n[latex]9\\left(8x - 3\\right)-\\left(-2\\right)[\/latex]\r\n\r\n[latex]4\\left(6x - 1\\right)-\\left(-8\\right)[\/latex]\r\n\r\n24x<\/em> + 4\r\n\r\n[latex]14\\left(c - 1\\right)-8\\left(c - 6\\right)[\/latex]\r\n\r\n[latex]11\\left(n - 7\\right)-5\\left(n - 1\\right)[\/latex]\r\n\r\n6n<\/em> \u2212 72\r\n\r\n[latex]6\\left(7y+8\\right)-\\left(30y - 15\\right)[\/latex]\r\n\r\n[latex]7\\left(3n+9\\right)-\\left(4n - 13\\right)[\/latex]\r\n\r\n17n<\/em> + 76\r\n

                              Evaluate Expressions Using the Distributive Property<\/strong><\/h3>\r\nIn the following exercises, evaluate both expressions for the given value.\r\n\r\nIf [latex]v=-2[\/latex], evaluate\r\n\r\n\u24d0 [latex]6\\left(4v+7\\right)[\/latex]\r\n\u24d1 [latex]6\\cdot 4v+6\\cdot 7[\/latex]\r\n\r\nIf [latex]u=-1[\/latex], evaluate\r\n\r\n\u24d0 [latex]8\\left(5u+12\\right)[\/latex]\r\n\u24d1 [latex]8\\cdot 5u+8\\cdot 12[\/latex]\r\n\r\n\u24d0 56\r\n\u24d1 56\r\n\r\nIf [latex]n=\\Large\\frac{2}{3}[\/latex], evaluate\r\n\r\n\u24d0 [latex]3\\left(n+\\Large\\frac{5}{6}\\normalsize\\right)[\/latex]\r\n\u24d1 [latex]3\\cdot n+3\\cdot\\Large\\frac{5}{6}[\/latex]\r\n\r\nIf [latex]y=\\Large\\frac{3}{4}[\/latex], evaluate\r\n\r\n\u24d0 [latex]4\\left(y+\\Large\\frac{3}{8}\\normalsize\\right)[\/latex]\r\n\u24d1 [latex]4\\cdot y+4\\cdot\\Large\\frac{3}{8}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{9}{2}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{9}{2}[\/latex]\r\n\r\nIf [latex]y=\\Large\\frac{7}{12}[\/latex], evaluate\r\n\r\n\u24d0 [latex]-3\\left(4y+15\\right)[\/latex]\r\n\u24d1 [latex]3\\cdot 4y+\\left(-3\\right)\\cdot 15[\/latex]\r\n\r\nIf [latex]p=\\Large\\frac{23}{30}[\/latex], evaluate\r\n\r\n\u24d0 [latex]-6\\left(5p+11\\right)[\/latex]\r\n\u24d1 [latex]-6\\cdot 5p+\\left(-6\\right)\\cdot 11[\/latex]\r\n\r\n\u24d0 \u221289\r\n\u24d1 \u221289\r\n\r\nIf [latex]m=0.4[\/latex], evaluate\r\n\r\n\u24d0 [latex]-10\\left(3m - 0.9\\right)[\/latex]\r\n\u24d1 [latex]-10\\cdot 3m-\\left(-10\\right)\\left(0.9\\right)[\/latex]\r\n\r\nIf [latex]n=0.75[\/latex], evaluate\r\n\r\n\u24d0 [latex]-100\\left(5n+1.5\\right)[\/latex]\r\n\u24d1 [latex]-100\\cdot 5n+\\left(-100\\right)\\left(1.5\\right)[\/latex]\r\n\r\n\u24d0 \u2212525\r\n\u24d1 \u2212525\r\n\r\nIf [latex]y=-25[\/latex], evaluate\r\n\r\n\u24d0 [latex]-\\left(y - 25\\right)[\/latex]\r\n\u24d1 [latex]-y+25[\/latex]\r\n\r\nIf [latex]w=-80[\/latex], evaluate\r\n\r\n\u24d0 [latex]-\\left(w - 80\\right)[\/latex]\r\n\u24d1 [latex]-w+80[\/latex]\r\n\r\n\u24d0 160\r\n\u24d1 160\r\n\r\nIf [latex]p=0.19[\/latex], evaluate\r\n\r\n\u24d0 [latex]-\\left(p+0.72\\right)[\/latex]\r\n\u24d1 [latex]-p - 0.72[\/latex]\r\n\r\nIf [latex]q=0.55[\/latex], evaluate\r\n\r\n\u24d0 [latex]-\\left(q+0.48\\right)[\/latex]\r\n\u24d1 [latex]-q - 0.48[\/latex]\r\n\r\n\u24d0 \u22121.03\r\n\u24d1 \u22121.03\r\n
                              \r\n

                              EVERYDAY MATH<\/h3>\r\n

                              Buying by the case<\/strong><\/h4>\r\nJoe can buy his favorite ice tea at a convenience store for [latex]{$1.99}[\/latex] per bottle. At the grocery store, he can buy a case of [latex]12[\/latex] bottles for [latex]{$23.88}[\/latex].\r\n\u24d0 Use the distributive property to find the cost of [latex]12[\/latex] bottles bought individually at the convenience store. (Hint: notice that [latex]{$1.99}[\/latex] is [latex]{$2}-{$0.01}[\/latex]. )\r\n\u24d1 Is it a bargain to buy the iced tea at the grocery store by the case?\r\n

                              Multi-pack purchase<\/strong><\/h4>\r\nAdele\u2019s shampoo sells for [latex]{$3.97}[\/latex] per bottle at the drug store. At the warehouse store, the same shampoo is sold as a [latex]{3-pack}[\/latex] for [latex]{$10.49}[\/latex].\r\n\u24d0 Show how you can use the distributive property to find the cost of [latex]3[\/latex] bottles bought individually at the drug store.\r\n\u24d1 How much would Adele save by buying the [latex]{3-pack}[\/latex] at the warehouse store?\r\n\r\n\u24d0 3(4 \u2212 0.03) = 11.91\r\n\u24d1 $1.42\r\n

                              <\/h2>\r\n<\/div>\r\n \r\n
                              \r\n

                              WRITING EXERCISES<\/h3>\r\nSimplify [latex]8\\left(x-\\Large\\frac{1}{4}\\normalsize\\right)[\/latex] using the distributive property and explain each step.\r\n\r\nExplain how you can multiply [latex]4\\left({$5.97}\\right)[\/latex] without paper or a calculator by thinking of [latex]{$5.97}[\/latex] as [latex]6 - 0.03[\/latex] and then using the distributive property.\r\n\r\n<\/div>\r\n \r\n

                              <\/h2>\r\n

                              Properties of Identity, Inverses, and Zero<\/h2>\r\nIn the following exercises, identify whether each example is using the identity property of addition or multiplication.\r\n\r\n[latex]101+0=101[\/latex]\r\n\r\n[latex]\\Large\\frac{3}{5}\\normalsize\\left(1\\right)=\\Large\\frac{3}{5}[\/latex]\r\n\r\nidentity property of multiplication\r\n\r\n[latex]-9\\cdot 1=-9[\/latex]\r\n\r\n[latex]0+64=64[\/latex]\r\n\r\nidentity property of addition\r\n\r\nUse the Inverse Properties of Addition and Multiplication<\/strong>\r\nIn the following exercises, find the multiplicative inverse.\r\n\r\n[latex]8[\/latex]\r\n\r\n[latex]14[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{14}[\/latex]\r\n\r\n[latex]-17[\/latex]\r\n\r\n[latex]-19[\/latex]\r\n\r\n[latex]-\\Large\\frac{1}{19}[\/latex]\r\n\r\n[latex]\\Large\\frac{7}{12}[\/latex]\r\n\r\n[latex]\\Large\\frac{8}{13}[\/latex]\r\n\r\n[latex]\\Large\\frac{13}{8}[\/latex]\r\n\r\n[latex]-\\Large\\frac{3}{10}[\/latex]\r\n\r\n[latex]-\\Large\\frac{5}{12}[\/latex]\r\n\r\n[latex]-\\Large\\frac{12}{5}[\/latex]\r\n\r\n[latex]0.8[\/latex]\r\n\r\n[latex]0.4[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{2}[\/latex]\r\n\r\n[latex]-0.2[\/latex]\r\n\r\n[latex]-0.5[\/latex]\r\n\r\n\u22122\r\n

                              Use the Properties of Zero<\/strong><\/h3>\r\nIn the following exercises, simplify using the properties of zero.\r\n\r\n[latex]48\\cdot 0[\/latex]\r\n\r\n[latex]\\Large\\frac{0}{6}[\/latex]\r\n\r\n0\r\n\r\n[latex]\\Large\\frac{3}{0}[\/latex]\r\n\r\n[latex]22\\cdot 0[\/latex]\r\n\r\n0\r\n\r\n[latex]0\\div\\Large\\frac{11}{12}[\/latex]\r\n\r\n[latex]\\Large\\frac{6}{0}[\/latex]\r\n\r\nundefined\r\n\r\n[latex]\\Large\\frac{0}{3}[\/latex]\r\n\r\n[latex]0\\div\\Large\\frac{7}{15}[\/latex]\r\n\r\n0\r\n\r\n[latex]0\\cdot\\Large\\frac{8}{15}[\/latex]\r\n\r\n[latex]\\left(-3.14\\right)\\left(0\\right)[\/latex]\r\n\r\n0\r\n\r\n[latex]5.72\\div 0[\/latex]\r\n\r\n[latex]\\Large\\frac{\\frac{1}{10}}{0}[\/latex]\r\n\r\nundefined\r\n

                              Simplify Expressions using the Properties of Identities, Inverses, and Zero<\/strong><\/h3>\r\nIn the following exercises, simplify using the properties of identities, inverses, and zero.\r\n\r\n[latex]19a+44 - 19a[\/latex]\r\n\r\n[latex]27c+16 - 27c[\/latex]\r\n\r\n16\r\n\r\n[latex]38+11r - 38[\/latex]\r\n\r\n[latex]92+31s - 92[\/latex]\r\n\r\n31s<\/em>\r\n\r\n[latex]10\\left(0.1d\\right)[\/latex]\r\n\r\n[latex]100\\left(0.01p\\right)[\/latex]\r\n\r\np<\/em>\r\n\r\n[latex]5\\left(0.6q\\right)[\/latex]\r\n\r\n[latex]40\\left(0.05n\\right)[\/latex]\r\n\r\n2n<\/em>\r\n\r\n[latex]\\Large\\frac{0}{r+20}[\/latex] , where [latex]r\\ne -20[\/latex]\r\n\r\n[latex]\\Large\\frac{0}{s+13}[\/latex] , where [latex]s\\ne -13[\/latex]\r\n\r\n0\r\n\r\n[latex]\\Large\\frac{0}{u - 4.99}[\/latex] , where [latex]u\\ne 4.99[\/latex]\r\n\r\n[latex]\\Large\\frac{0}{v - 65.1}[\/latex] , where [latex]v\\ne 65.1[\/latex]\r\n\r\n0\r\n\r\n[latex]0\\div \\left(x-\\Large\\frac{1}{2}\\normalsize\\right)[\/latex] , where [latex]x\\ne\\Large\\frac{1}{2}[\/latex]\r\n\r\n[latex]0\\div \\left(y-\\Large\\frac{1}{6}\\normalsize\\right)[\/latex] , where [latex]y\\ne\\Large\\frac{1}{6}[\/latex]\r\n\r\n0\r\n\r\n[latex]\\Large\\frac{32 - 5a}{0}[\/latex] , where [latex]32 - 5a\\ne 0[\/latex]\r\n\r\n[latex]\\Large\\frac{28 - 9b}{0}[\/latex] , where [latex]28 - 9b\\ne 0[\/latex]\r\n\r\nundefined\r\n\r\n[latex]\\Large\\frac{2.1+0.4c}{0}[\/latex] , where [latex]2.1+0.4c\\ne 0[\/latex]\r\n\r\n[latex]\\Large\\frac{1.75+9f}{0}[\/latex] , where [latex]1.75+9f\\ne 0[\/latex]\r\n\r\nundefined\r\n\r\n[latex]\\left(\\Large\\frac{3}{4}\\normalsize +\\Large\\frac{9}{10}\\normalsize m\\right)\\div 0[\/latex] , where [latex]\\Large\\frac{3}{4}\\normalsize +\\Large\\frac{9}{10}\\normalsize m\\ne 0[\/latex]\r\n\r\n[latex]\\left(\\Large\\frac{5}{16}\\normalsize n-\\Large\\frac{3}{7}\\normalsize\\right)\\div 0[\/latex] , where [latex]\\Large\\frac{5}{16}\\normalsize n-\\Large\\frac{3}{7}\\normalsize \\ne 0[\/latex]\r\n\r\nundefined\r\n\r\n[latex]\\Large\\frac{9}{10}\\normalsize\\cdot\\Large\\frac{10}{9}\\normalsize\\left(18p - 21\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{7}\\normalsize\\cdot\\Large\\frac{7}{5}\\normalsize\\left(20q - 35\\right)[\/latex]\r\n\r\n20q<\/em> \u2212 35\r\n\r\n[latex]15\\cdot\\Large\\frac{3}{5}\\normalsize\\left(4d+10\\right)[\/latex]\r\n\r\n[latex]18\\cdot\\Large\\frac{5}{6}\\normalsize\\left(15h+24\\right)[\/latex]\r\n\r\n225h<\/em> + 360\r\n
                              \r\n

                              EVERYDAY MATH<\/h3>\r\n

                              Insurance copayment<\/strong><\/h4>\r\nCarrie had to have [latex]5[\/latex] fillings done. Each filling cost [latex]{$80}[\/latex]. Her dental insurance required her to pay [latex]20%[\/latex] of the cost. Calculate Carrie\u2019s cost\r\n\u24d0 by finding her copay for each filling, then finding her total cost for [latex]5[\/latex] fillings, and\r\n\u24d1 by multiplying [latex]5\\left(0.20\\right)\\left(80\\right)[\/latex].\r\n\u24d2 Which of the Properties of Real Numbers did you use for part (b)?\r\n

                              Cooking time<\/strong><\/h4>\r\nHelen bought a [latex]\\text{24-pound}[\/latex] turkey for her family\u2019s Thanksgiving dinner and wants to know what time to put the turkey in the oven. She wants to allow [latex]20[\/latex] minutes per pound cooking time.\r\n\u24d0 Calculate the length of time needed to roast the turkey by multiplying [latex]24\\cdot 20[\/latex] to find the number of minutes and then multiplying the product by [latex]\\Large\\frac{1}{60}[\/latex] to convert minutes into hours.\r\n\u24d1 Multiply [latex]24\\left(20\\cdot\\Large\\frac{1}{60}\\normalsize\\right)[\/latex].\r\n\u24d2 Which of the Properties of Real Numbers allows you to multiply [latex]24\\left(20\\cdot\\Large\\frac{1}{60}\\right)[\/latex] instead of [latex]\\left(24\\cdot 20\\right)\\Large\\frac{1}{60}[\/latex]?\r\n\r\n\u24d0 8 hours\r\n\u24d1 8\r\n\u24d2 associative property of multiplication\r\n

                              <\/h2>\r\n<\/div>\r\n \r\n
                              \r\n

                              WRITING EXERCISES<\/h3>\r\nIn your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.\r\n\r\nHow can the use of the properties of real numbers make it easier to simplify expressions?\r\n\r\nAnswers will vary.\r\n\r\n<\/div>\r\n \r\n

                              Determine Whether a Number is a Solution of an Equation<\/strong><\/h2>\r\nIn the following exercises, determine whether each number is a solution of the given equation.\r\n\r\n[latex]4x - 2=6[\/latex]\r\n\r\n\u24d0 [latex]x=-2[\/latex]\r\n\u24d1 [latex]x=-1[\/latex]\r\n\u24d2 [latex]x=2[\/latex]\r\n
                                \r\n \t
                              1. \u24d0 no<\/li>\r\n \t
                              2. \u24d1 yes<\/li>\r\n \t
                              3. \u24d2 no<\/li>\r\n<\/ol>\r\n[latex]4y - 10=-14[\/latex]\r\n\r\n\u24d0 [latex]y=-6[\/latex]\r\n\u24d1 [latex]y=-1[\/latex]\r\n\u24d2 [latex]y=1[\/latex]\r\n\r\n[latex]9a+27=-63[\/latex]\r\n\r\n\u24d0 [latex]a=6[\/latex]\r\n\u24d1 [latex]a=-6[\/latex]\r\n\u24d2 [latex]a=-10[\/latex]\r\n
                                  \r\n \t
                                1. \u24d0 no<\/li>\r\n \t
                                2. \u24d1 no<\/li>\r\n \t
                                3. \u24d2 yes<\/li>\r\n<\/ol>\r\n[latex]7c+42=-56[\/latex]\r\n\r\n\u24d0 [latex]c=2[\/latex]\r\n\u24d1 [latex]c=-2[\/latex]\r\n\u24d2 [latex]c=-14[\/latex]\r\n

                                  Solve Equations Using the Addition and Subtraction Properties of Equality<\/strong><\/h3>\r\nIn the following exercises, solve for the unknown.\r\n\r\n[latex]n+12=5[\/latex]\r\n\r\n\u22127\r\n\r\n[latex]m+16=2[\/latex]\r\n\r\n[latex]p+9=-8[\/latex]\r\n\r\n\u221217\r\n\r\n[latex]q+5=-6[\/latex]\r\n\r\n[latex]u - 3=-7[\/latex]\r\n\r\n\u22124\r\n\r\n[latex]v - 7=-8[\/latex]\r\n\r\n[latex]h - 10=-4[\/latex]\r\n\r\n6\r\n\r\n[latex]k - 9=-5[\/latex]\r\n\r\n[latex]x+\\left(-2\\right)=-18[\/latex]\r\n\r\n\u221216\r\n\r\n[latex]y+\\left(-3\\right)=-10[\/latex]\r\n\r\n[latex]r-\\left(-5\\right)=-9[\/latex]\r\n\r\n\u221214\r\n\r\n[latex]s-\\left(-2\\right)=-11[\/latex]\r\n

                                  Model the Division Property of Equality<\/strong><\/h3>\r\nIn the following exercises, write the equation modeled by the envelopes and counters and then solve it.\r\n\r\n\"This\r\n\r\n3x<\/em> = 6; x<\/em> = 2\r\n\r\n\"This\r\n\r\n\"This\r\n\r\n2x<\/em> = 8; x<\/em> = 4\r\n\r\n\"This\r\n

                                  Solve Equations Using the Division Property of Equality<\/strong><\/h3>\r\nIn the following exercises, solve each equation using the division property of equality and check the solution.\r\n\r\n[latex]5x=45[\/latex]\r\n\r\n9\r\n\r\n[latex]4p=64[\/latex]\r\n\r\n[latex]-7c=56[\/latex]\r\n\r\n\u22128\r\n\r\n[latex]-9x=54[\/latex]\r\n\r\n[latex]-14p=-42[\/latex]\r\n\r\n3\r\n\r\n[latex]-8m=-40[\/latex]\r\n\r\n[latex]-120=10q[\/latex]\r\n\r\n\u221212\r\n\r\n[latex]-75=15y[\/latex]\r\n\r\n[latex]24x=480[\/latex]\r\n\r\n20\r\n\r\n[latex]18n=540[\/latex]\r\n\r\n[latex]-3z=0[\/latex]\r\n\r\n0\r\n\r\n[latex]4u=0[\/latex]\r\n

                                  Translate to an Equation and Solve<\/strong><\/h3>\r\nIn the following exercises, translate and solve.\r\n\r\nFour more than [latex]n[\/latex] is equal to 1.\r\n\r\nn<\/em> + 4 = 1; n<\/em> = \u22123\r\n\r\nNine more than [latex]m[\/latex] is equal to 5.\r\n\r\nThe sum of eight and [latex]p[\/latex] is [latex]-3[\/latex] .\r\n\r\n8 + p<\/em> = \u22123; p<\/em> = \u221211\r\n\r\nThe sum of two and [latex]q[\/latex] is [latex]-7[\/latex] .\r\n\r\nThe difference of [latex]a[\/latex] and three is [latex]-14[\/latex] .\r\n\r\na<\/em> \u2212 3 = \u221214; a<\/em> = \u221211\r\n\r\nThe difference of [latex]b[\/latex] and [latex]5[\/latex] is [latex]-2[\/latex] .\r\n\r\nThe number \u221242 is the product of \u22127 and [latex]x[\/latex] .\r\n\r\n\u221242 = \u22127x<\/em>; x<\/em> = 6\r\n\r\nThe number \u221254 is the product of \u22129 and [latex]y[\/latex] .\r\n\r\nThe product of [latex]f[\/latex] and \u221215 is 75.\r\n\r\nf<\/em>(\u221215) = 75; f<\/em> = 5\r\n\r\nThe product of [latex]g[\/latex] and \u221218 is 36.\r\n\r\n\u22126 plus [latex]c[\/latex] is equal to 4.\r\n\r\n\u22126 + c<\/em> = 4; c<\/em> = 10\r\n\r\n\u22122 plus [latex]d[\/latex] is equal to 1.\r\n\r\nNine less than [latex]n[\/latex] is \u22124.\r\n\r\nm<\/em> \u2212 9 = \u22124; m<\/em> = 5\r\n\r\nThirteen less than [latex]n[\/latex] is [latex]-10[\/latex] .\r\n

                                  Mixed Practice<\/strong><\/h3>\r\nIn the following exercises, solve.\r\n
                                    \r\n \t
                                  1. \u24d0 [latex]x+2=10[\/latex]<\/li>\r\n \t
                                  2. \u24d1 [latex]2x=10[\/latex]<\/li>\r\n<\/ol>\r\n
                                      \r\n \t
                                    1. \u24d0 8<\/li>\r\n \t
                                    2. \u24d1 5<\/li>\r\n<\/ol>\r\n
                                        \r\n \t
                                      1. \u24d0 [latex]y+6=12[\/latex]<\/li>\r\n \t
                                      2. \u24d1 [latex]6y=12[\/latex]<\/li>\r\n<\/ol>\r\n
                                          \r\n \t
                                        1. \u24d0 [latex]-3p=27[\/latex]<\/li>\r\n \t
                                        2. \u24d1 [latex]p - 3=27[\/latex]<\/li>\r\n<\/ol>\r\n
                                            \r\n \t
                                          1. \u24d0 \u22129<\/li>\r\n \t
                                          2. \u24d1 30<\/li>\r\n<\/ol>\r\n
                                              \r\n \t
                                            1. \u24d0 [latex]-2q=34[\/latex]<\/li>\r\n \t
                                            2. \u24d1 [latex]q - 2=34[\/latex]<\/li>\r\n<\/ol>\r\n[latex]a - 4=16[\/latex]\r\n\r\n20\r\n\r\n[latex]b - 1=11[\/latex]\r\n\r\n[latex]-8m=-56[\/latex]\r\n\r\n7\r\n\r\n[latex]-6n=-48[\/latex]\r\n\r\n[latex]-39=u+13[\/latex]\r\n\r\n\u221252\r\n\r\n[latex]-100=v+25[\/latex]\r\n\r\n[latex]11r=-99[\/latex]\r\n\r\n\u22129\r\n\r\n[latex]15s=-300[\/latex]\r\n\r\n[latex]100=20d[\/latex]\r\n\r\n5\r\n\r\n[latex]250=25n[\/latex]\r\n\r\n[latex]-49=x - 7[\/latex]\r\n\r\n\u221242\r\n\r\n[latex]64=y - 4[\/latex]\r\n
                                              \r\n

                                              EVERYDAY MATH<\/h3>\r\n

                                              Cookie packaging<\/strong><\/h4>\r\nA package of [latex]\\text{51 cookies}[\/latex] has [latex]3[\/latex] equal rows of cookies. Find the number of cookies in each row, [latex]c[\/latex], by solving the equation [latex]3c=51[\/latex].\r\n\r\n17 cookies\r\n

                                              Kindergarten class<\/strong><\/h4>\r\nConnie\u2019s kindergarten class has [latex]\\text{24 children.}[\/latex] She wants them to get into [latex]4[\/latex] equal groups. Find the number of children in each group, [latex]g[\/latex], by solving the equation [latex]4g=24[\/latex].\r\n

                                              <\/h2>\r\n<\/div>\r\n \r\n
                                              \r\n

                                              WRITING EXERCISES<\/h3>\r\nIs modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation [latex]3x=15?[\/latex] Explain why or why not.\r\n\r\nSample answer: It is helpful because it shows how the counters can be divided among the envelopes.\r\n\r\nSuppose you are using envelopes and counters to model solving the equations [latex]x+4=12[\/latex] and [latex]4x=12[\/latex]. Explain how you would solve each equation.\r\n\r\nFrida started to solve the equation [latex]-3x=36[\/latex] by adding [latex]3[\/latex] to both sides. Explain why Frida\u2019s method will not solve the equation.\r\n\r\nSample answer: The operation used in the equation is multiplication. The inverse of multiplication is division, not addition.\r\n\r\nRaoul started to solve the equation [latex]4y=40[\/latex] by subtracting [latex]4[\/latex] from both sides. Explain why Raoul\u2019s method will not solve the equation.\r\n

                                              <\/h2>\r\n<\/div>\r\n \r\n

                                              Chapter Review Exercises<\/h3>\r\n
                                              \r\n

                                              Introduction to integers<\/h3>\r\n

                                              Locate Positive and Negative Numbers on the Number Line<\/strong><\/h4>\r\nIn the following exercises, locate and label the integer on the number line.\r\n\r\n[latex]5[\/latex]\r\n\r\n\"This\r\n\r\n[latex]-5[\/latex]\r\n\r\n[latex]-3[\/latex]\r\n\r\n\"This\r\n\r\n[latex]3[\/latex]\r\n\r\n[latex]-8[\/latex]\r\n\r\n\"This\r\n\r\n[latex]-7[\/latex]\r\n

                                              Order Positive and Negative Numbers<\/strong><\/h4>\r\nIn the following exercises, order each of the following pairs of numbers, using [latex]&lt[\/latex]; or [latex]\\text{>.}[\/latex]\r\n\r\n[latex]4\\text{__}8[\/latex]\r\n\r\n<\r\n\r\n[latex]-6\\text{__}3[\/latex]\r\n\r\n[latex]-5\\text{__}-10[\/latex]\r\n\r\n>\r\n\r\n[latex]-9\\text{__}-4[\/latex]\r\n\r\n[latex]2\\text{__}-7[\/latex]\r\n\r\n>\r\n\r\n[latex]-3\\text{__}1[\/latex]\r\n

                                              Find Opposites<\/strong><\/h4>\r\nIn the following exercises, find the opposite of each number.\r\n\r\n[latex]6[\/latex]\r\n\r\n\u22126\r\n\r\n[latex]-2[\/latex]\r\n\r\n[latex]-4[\/latex]\r\n\r\n4\r\n\r\n[latex]3[\/latex]\r\n\r\nIn the following exercises, simplify.\r\n
                                                \r\n \t
                                              1. \u24d0 [latex]-\\left(8\\right)[\/latex]<\/li>\r\n \t
                                              2. \u24d1 [latex]-\\left(-8\\right)[\/latex]<\/li>\r\n<\/ol>\r\n
                                                  \r\n \t
                                                1. \u24d0 \u22128<\/li>\r\n \t
                                                2. \u24d1 8<\/li>\r\n<\/ol>\r\n
                                                    \r\n \t
                                                  1. \u24d0 [latex]-\\left(9\\right)[\/latex]<\/li>\r\n \t
                                                  2. \u24d1 [latex]-\\left(-9\\right)[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercises, evaluate.\r\n\r\n[latex]-x,\\text{when}[\/latex]\r\n\r\n\u24d0 [latex]x=32[\/latex]\r\n\u24d1 [latex]x=-32[\/latex]\r\n
                                                      \r\n \t
                                                    1. \u24d0 \u221232<\/li>\r\n \t
                                                    2. \u24d1 32<\/li>\r\n<\/ol>\r\n[latex]-n,\\text{when}[\/latex]\r\n\r\n\u24d0 [latex]n=20[\/latex]\r\n\u24d0 [latex]n=-20[\/latex]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                      \r\n

                                                      ABSOLUTE VALUES<\/h3>\r\n

                                                      Simplify Absolute Values<\/strong><\/h4>\r\nIn the following exercises, simplify.\r\n\r\n[latex]|-21|[\/latex]\r\n\r\n21\r\n\r\n[latex]|-42|[\/latex]\r\n\r\n[latex]|36|[\/latex]\r\n\r\n36\r\n\r\n[latex]-|15|[\/latex]\r\n\r\n[latex]|0|[\/latex]\r\n\r\n0\r\n\r\n[latex]-|-75|[\/latex]\r\n\r\nIn the following exercises, evaluate.\r\n\r\n[latex]|x|\\text{when}x=-14[\/latex]\r\n\r\n14\r\n\r\n[latex]-|r|\\text{when}r=27[\/latex]\r\n\r\n[latex]-|-y|\\text{when}y=33[\/latex]\r\n\r\n\u221233\r\n\r\n[latex]|{\\text{-n}}|\\text{when}n=-4[\/latex]\r\n\r\nIn the following exercises, fill in [latex]\\text{<},\\text{>},\\text{or}=[\/latex] for each of the following pairs of numbers.\r\n\r\n[latex]-|-4|\\text{__}4[\/latex]\r\n\r\n<\r\n\r\n[latex]-2\\text{__}|-2|[\/latex]\r\n\r\n[latex]-|-6|\\text{__}-6[\/latex]\r\n\r\n=\r\n\r\n[latex]-|-9|\\text{__}|-9|[\/latex]\r\n\r\nIn the following exercises, simplify.\r\n\r\n[latex]-\\left(-55\\right)\\text{and}-|-55|[\/latex]\r\n\r\n\u221255; \u221255\r\n\r\n[latex]-\\left(-48\\right)\\text{and}-|-48|[\/latex]\r\n\r\n[latex]|12 - 5|[\/latex]\r\n\r\n7\r\n\r\n[latex]|9+7|[\/latex]\r\n\r\n[latex]6|-9|[\/latex]\r\n\r\n54\r\n\r\n[latex]|14 - 8|-|-2|[\/latex]\r\n\r\n[latex]|9 - 3|-|5 - 12|[\/latex]\r\n\r\n\u22121\r\n\r\n[latex]5+4|15 - 3|[\/latex]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                      \r\n

                                                      Translate Phrases to Expressions with Integers<\/strong><\/h3>\r\nIn the following exercises, translate each of the following phrases into expressions with positive or negative numbers.\r\n\r\nthe opposite of [latex]16[\/latex]\r\n\r\n\u221216\r\n\r\nthe opposite of [latex]-8[\/latex]\r\n\r\nnegative [latex]3[\/latex]\r\n\r\n\u22123\r\n\r\n[latex]19[\/latex] minus negative [latex]12[\/latex]\r\n\r\na temperature of [latex]10[\/latex] below zero\r\n\r\n\u221210\u00b0\r\n\r\nan elevation of [latex]\\text{85 feet}[\/latex] below sea level\r\n\r\nAdd Integers\r\n

                                                      Model Addition of Integers<\/strong><\/h4>\r\nIn the following exercises, model the following to find the sum.\r\n\r\n[latex]3+7[\/latex]\r\n\r\n10\r\n\r\n[latex]-2+6[\/latex]\r\n\r\n[latex]5+\\left(-4\\right)[\/latex]\r\n\r\n1\r\n\r\n[latex]-3+\\left(-6\\right)[\/latex]\r\n

                                                      Simplify Expressions with Integers<\/strong><\/h4>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]14+82[\/latex]\r\n\r\n96\r\n\r\n[latex]-33+\\left(-67\\right)[\/latex]\r\n\r\n[latex]-75+25[\/latex]\r\n\r\n\u221250\r\n\r\n[latex]54+\\left(-28\\right)[\/latex]\r\n\r\n[latex]11+\\left(-15\\right)+3[\/latex]\r\n\r\n\u22121\r\n\r\n[latex]-19+\\left(-42\\right)+12[\/latex]\r\n\r\n[latex]-3+6\\left(-1+5\\right)[\/latex]\r\n\r\n21\r\n\r\n[latex]10+4\\left(-3+7\\right)[\/latex]\r\n

                                                      Evaluate Variable Expressions with Integers<\/strong><\/h4>\r\nIn the following exercises, evaluate each expression.\r\n\r\n[latex]n+4\\text{when}[\/latex]\r\n\r\n\u24d0 [latex]n=-1[\/latex]\r\n\u24d1 [latex]n=-20[\/latex]\r\n
                                                        \r\n \t
                                                      1. \u24d0 3<\/li>\r\n \t
                                                      2. \u24d1 \u221216<\/li>\r\n<\/ol>\r\n[latex]x+\\left(-9\\right)\\text{when}[\/latex]\r\n\r\n\u24d0 [latex]x=3[\/latex]\r\n\u24d1 [latex]x=-3[\/latex]\r\n\r\n[latex]{\\left(x+y\\right)}^{3}\\text{when}x=-4,y=1[\/latex]\r\n\r\n\u221227\r\n\r\n[latex]{\\left(u+v\\right)}^{2}\\text{when}u=-4,v=11[\/latex]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                        \r\n

                                                        TRANSLATE WORD PHRASES TO ALGEBRAIC EXPRESSIONS<\/h3>\r\nIn the following exercises, translate each phrase into an algebraic expression and then simplify.\r\n\r\n[latex]\\text{the sum of -8 and 2}[\/latex]\r\n\r\n\u22128 + 2 = \u22126\r\n\r\n[latex]\\text{4 more than -12}[\/latex]\r\n\r\n[latex]\\text{10 more than the sum of -5 and -6}[\/latex]\r\n\r\n10 + [\u22125 + (\u22126)] = \u22121\r\n\r\n[latex]\\text{the sum of}3\\text{and}-5,\\text{increased by 18}[\/latex]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                        \r\n

                                                        Add Integers in Applications<\/strong><\/h3>\r\nIn the following exercises, solve.\r\n

                                                        Temperature<\/strong><\/h4>\r\nOn Monday, the high temperature in Denver was [latex]\\text{-4 degrees.}[\/latex] Tuesday\u2019s high temperature was [latex]\\text{20 degrees}[\/latex] more. What was the high temperature on Tuesday?\r\n\r\n16 degrees\r\n

                                                        Credit<\/strong><\/h4>\r\nFrida owed [latex]{$75}[\/latex] on her credit card. Then she charged [latex]{$21}[\/latex] more. What was her new balance?\r\n\r\nSubtract Integers\r\n

                                                        Model Subtraction of Integers<\/strong><\/h4>\r\nIn the following exercises, model the following.\r\n\r\n[latex]6 - 1[\/latex]\r\n\r\n\"This\r\n\r\n5\r\n\r\n[latex]-4-\\left(-3\\right)[\/latex]\r\n\r\n[latex]2-\\left(-5\\right)[\/latex]\r\n\r\n\"This\r\n\r\n7\r\n\r\n[latex]-1 - 4[\/latex]\r\n

                                                        Simplify Expressions with Integers<\/strong><\/h4>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]24 - 16[\/latex]\r\n\r\n8\r\n\r\n[latex]19-\\left(-9\\right)[\/latex]\r\n\r\n[latex]-31 - 7[\/latex]\r\n\r\n\u221238\r\n\r\n[latex]-40-\\left(-11\\right)[\/latex]\r\n\r\n[latex]-52-\\left(-17\\right)-23[\/latex]\r\n\r\n\u221258\r\n\r\n[latex]25-\\left(-3 - 9\\right)[\/latex]\r\n\r\n[latex]\\left(1 - 7\\right)-\\left(3 - 8\\right)[\/latex]\r\n\r\n\u22121\r\n\r\n[latex]{3}^{2}-{7}^{2}[\/latex]\r\n\r\nEvaluate Variable Expressions with Integers<\/strong>\r\nIn the following exercises, evaluate each expression.\r\n\r\n[latex]x - 7\\text{when}[\/latex]\r\n\r\n\u24d0 [latex]x=5[\/latex]\r\n\u24d1 [latex]x=-4[\/latex]\r\n
                                                          \r\n \t
                                                        1. \u24d0 \u22122<\/li>\r\n \t
                                                        2. \u24d1 \u221211<\/li>\r\n<\/ol>\r\n[latex]10-y\\text{when}[\/latex]\r\n\r\n\u24d0 [latex]y=15[\/latex]\r\n\u24d1 [latex]y=-16[\/latex]\r\n\r\n[latex]2{n}^{2}-n+5\\text{when}n=-4[\/latex]\r\n\r\n41\r\n\r\n[latex]-15 - 3{u}^{2}\\text{when}u=-5[\/latex]\r\n

                                                          Translate Phrases to Algebraic Expressions<\/strong><\/h4>\r\nIn the following exercises, translate each phrase into an algebraic expression and then simplify.\r\n\r\nthe difference of [latex]-12\\text{and}5[\/latex]\r\n\r\n\u221212 \u2212 5 = \u221217\r\n\r\nsubtract [latex]23[\/latex] from [latex]-50[\/latex]\r\n

                                                          Subtract Integers in Applications<\/strong><\/h4>\r\nIn the following exercises, solve the given applications.\r\n

                                                          Temperature<\/strong><\/h4>\r\nOne morning the temperature in Bangor, Maine was [latex]\\text{18 degrees.}[\/latex] By afternoon, it had dropped [latex]\\text{20 degrees.}[\/latex] What was the afternoon temperature?\r\n\r\n\u22122 degrees\r\n

                                                          Temperature<\/strong><\/h4>\r\nOn January 4, the high temperature in Laredo, Texas was [latex]\\text{78 degrees,}[\/latex] and the high in Houlton, Maine was [latex]-28\\text{degrees}[\/latex]. What was the difference in temperature of Laredo and Houlton?\r\n\r\n \r\n\r\n<\/div>\r\n
                                                          \r\n

                                                          Multiply and Divide Integers<\/h3>\r\nMultiply Integers<\/strong>\r\nIn the following exercises, multiply.\r\n\r\n[latex]-9\\cdot 4[\/latex]\r\n\r\n\u221236\r\n\r\n[latex]5\\left(-7\\right)[\/latex]\r\n\r\n[latex]\\left(-11\\right)\\left(-11\\right)[\/latex]\r\n\r\n121\r\n\r\n[latex]-1\\cdot 6[\/latex]\r\n\r\nDivide Integers<\/strong>\r\nIn the following exercises, divide.\r\n\r\n[latex]56\\div \\left(-8\\right)[\/latex]\r\n\r\n\u22127\r\n\r\n[latex]-120\\div \\left(-6\\right)[\/latex]\r\n\r\n[latex]-96\\div 12[\/latex]\r\n\r\n\u22128\r\n\r\n[latex]96\\div \\left(-16\\right)[\/latex]\r\n\r\n[latex]45\\div \\left(-1\\right)[\/latex]\r\n\r\n\u221245\r\n\r\n[latex]-162\\div \\left(-1\\right)[\/latex]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                          \r\n

                                                          Simplify Expressions with Integers<\/strong><\/h3>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]5\\left(-9\\right)-3\\left(-12\\right)[\/latex]\r\n\r\n\u22129\r\n\r\n[latex]{\\left(-2\\right)}^{5}[\/latex]\r\n\r\n[latex]-{3}^{4}[\/latex]\r\n\r\n\u221281\r\n\r\n[latex]\\left(-3\\right)\\left(4\\right)\\left(-5\\right)\\left(-6\\right)[\/latex]\r\n\r\n[latex]42 - 4\\left(6 - 9\\right)[\/latex]\r\n\r\n54\r\n\r\n[latex]\\left(8 - 15\\right)\\left(9 - 3\\right)[\/latex]\r\n\r\n[latex]-2\\left(-18\\right)\\div 9[\/latex]\r\n\r\n4\r\n\r\n[latex]45\\div \\left(-3\\right)-12[\/latex]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                          \r\n

                                                          Evaluate Variable Expressions with Integers<\/strong><\/h3>\r\nIn the following exercises, evaluate each expression.\r\n\r\n[latex]7x - 3\\text{when}x=-9[\/latex]\r\n\r\n\u221266\r\n\r\n[latex]16 - 2n\\text{when}n=-8[\/latex]\r\n\r\n[latex]5a+8b\\text{when}a=-2,b=-6[\/latex]\r\n\r\n\u221258\r\n\r\n[latex]{x}^{2}+5x+4\\text{when}x=-3[\/latex]\r\n\r\n \r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                          \r\n

                                                          Translate Word Phrases to Algebraic Expressions<\/strong><\/h3>\r\nIn the following exercises, translate to an algebraic expression and simplify if possible.\r\n\r\nthe product of [latex]-12[\/latex] and [latex]6[\/latex]\r\n\r\n\u221212(6) = \u221272\r\n\r\nthe quotient of [latex]3[\/latex] and the sum of [latex]-7[\/latex] and [latex]s[\/latex]\r\n\r\nSolve Equations using Integers; The Division Property of Equality\r\nDetermine Whether a Number is a Solution of an Equation<\/strong>\r\nIn the following exercises, determine whether each number is a solution of the given equation.\r\n\r\n[latex]5x - 10=-35[\/latex]\r\n\r\n\u24d0 [latex]x=-9[\/latex]\r\n\u24d1 [latex]x=-5[\/latex]\r\n\u24d2 [latex]x=5[\/latex]\r\n
                                                            \r\n \t
                                                          1. \u24d0 no<\/li>\r\n \t
                                                          2. \u24d1 yes<\/li>\r\n \t
                                                          3. \u24d2 no<\/li>\r\n<\/ol>\r\n[latex]8u+24=-32[\/latex]\r\n\r\n\u24d0 [latex]u=-7[\/latex]\r\n\u24d1 [latex]u=-1[\/latex]\r\n\u24d2 [latex]u=7[\/latex]\r\n\r\nUsing the Addition and Subtraction Properties of Equality<\/strong>\r\nIn the following exercises, solve.\r\n\r\n[latex]a+14=2[\/latex]\r\n\r\n\u221212\r\n\r\n[latex]b - 9=-15[\/latex]\r\n\r\n\u22126\r\n\r\n[latex]c+\\left(-10\\right)=-17[\/latex]\r\n\r\n\u22127\r\n\r\n[latex]d-\\left(-6\\right)=-26[\/latex]\r\n\r\n\u221232\r\n\r\n \r\n\r\n<\/div>\r\n \r\n
                                                            \r\n

                                                            Model the Division Property of Equality<\/strong><\/h3>\r\nIn the following exercises, write the equation modeled by the envelopes and counters. Then solve it.\r\n\r\n\"This\r\n\r\n3x<\/em> = 9; x<\/em> = 3>\r\n\r\n\"This\r\n\r\nSolve Equations Using the Division Property of Equality<\/strong>\r\nIn the following exercises, solve each equation using the division property of equality and check the solution.\r\n\r\n[latex]8p=72[\/latex]\r\n\r\n9\r\n\r\n[latex]-12q=48[\/latex]\r\n\r\n[latex]-16r=-64[\/latex]\r\n\r\n4\r\n\r\n[latex]-5s=-100[\/latex]\r\n\r\nTranslate to an Equation and Solve.<\/strong>\r\nIn the following exercises, translate and solve.\r\n\r\n[latex]\\text{The product of -6 and}y\\text{is}-42[\/latex]\r\n\r\n\u22126y<\/em> = \u221242; y<\/em> = 7\r\n\r\n[latex]\\text{The difference of}z\\text{and -13 is -18.}[\/latex]\r\n\r\nFour more than [latex]m[\/latex] is [latex]-48[\/latex].\r\n\r\nm<\/em> + 4 = \u221248; m<\/em> = \u221252\r\n\r\n[latex]\\text{The product of -21 and}n\\text{is 63.}[\/latex]\r\n

                                                            <\/h2>\r\n<\/div>\r\n \r\n
                                                            \r\n

                                                            Everyday Math<\/h3>\r\nDescribe how you have used two topics from this chapter in your life outside of your math class during the past month.\r\n\r\n<\/div>\r\n \r\n
                                                            \r\n

                                                            Chapter Practice Test<\/h3>\r\nLocate and label [latex]0,2,-4[\/latex], and [latex]-1[\/latex] on a number line.\r\n\r\nIn the following exercises, compare the numbers, using [latex]<\\text{or}>\\text{or}\\text{=.}[\/latex]\r\n
                                                              \r\n \t
                                                            1. \u24d0 [latex]-6\\text{__}3[\/latex]<\/li>\r\n \t
                                                            2. \u24d1 [latex]-1\\text{__}-4[\/latex]<\/li>\r\n<\/ol>\r\n
                                                                \r\n \t
                                                              1. \u24d0 <<\/li>\r\n \t
                                                              2. \u24d1 ><\/li>\r\n<\/ol>\r\n
                                                                  \r\n \t
                                                                1. \u24d0 [latex]-5\\text{__}|-5|[\/latex]<\/li>\r\n \t
                                                                2. \u24d1 [latex]-|-2|\\text{__}-2[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercises, find the opposite of each number.\r\n
                                                                    \r\n \t
                                                                  1. \u24d0 [latex]-7[\/latex]<\/li>\r\n \t
                                                                  2. \u24d1 [latex]8[\/latex]<\/li>\r\n<\/ol>\r\n
                                                                      \r\n \t
                                                                    1. \u24d0 7<\/li>\r\n \t
                                                                    2. \u24d1 \u22128<\/li>\r\n<\/ol>\r\nIn the following exercises, simplify.\r\n\r\n[latex]-\\left(-22\\right)[\/latex]\r\n\r\n[latex]|4 - 9|[\/latex]\r\n\r\n5\r\n\r\n[latex]-8+6[\/latex]\r\n\r\n[latex]-15+\\left(-12\\right)[\/latex]\r\n\r\n\u221227\r\n\r\n[latex]-7-\\left(-3\\right)[\/latex]\r\n\r\n[latex]10-\\left(5 - 6\\right)[\/latex]\r\n\r\n11\r\n\r\n[latex]-3\\cdot 8[\/latex]\r\n\r\n[latex]-6\\left(-9\\right)[\/latex]\r\n\r\n54\r\n\r\n[latex]70\\div \\left(-7\\right)[\/latex]\r\n\r\n[latex]{\\left(-2\\right)}^{3}[\/latex]\r\n\r\n\u22128\r\n\r\n[latex]-{4}^{2}[\/latex]\r\n\r\n[latex]16 - 3\\left(5 - 7\\right)[\/latex]\r\n\r\n22\r\n\r\n[latex]|21 - 6|-|-8|[\/latex]\r\n\r\nIn the following exercises, evaluate.\r\n\r\n[latex]35-a\\text{when}a=-4[\/latex]\r\n\r\n39\r\n\r\n[latex]{\\left(-2r\\right)}^{2}\\text{when}r=3[\/latex]\r\n\r\n[latex]3m - 2n\\text{when}m=6,n=-8[\/latex]\r\n\r\n34\r\n\r\n[latex]-|-y|\\text{when}y=17[\/latex]\r\n\r\nIn the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.\r\n\r\nthe difference of \u22127 and \u22124\r\n\r\n\u22127 \u2212 (\u22124) = \u22123\r\n\r\nthe quotient of [latex]25[\/latex] and the sum of [latex]m[\/latex] and [latex]n[\/latex].\r\n\r\nIn the following exercises, solve.\r\n\r\nEarly one morning, the temperature in Syracuse was [latex]-8\\text{^\\circ F.}[\/latex] By noon, it had risen [latex]\\text{12^\\circ .}[\/latex] What was the temperature at noon?\r\n\r\n4\u00b0F\r\n\r\nCollette owed [latex]{$128}[\/latex] on her credit card. Then she charged [latex]{$65.}[\/latex] What was her new balance?\r\n\r\nIn the following exercises, solve.\r\n\r\n[latex]n+6=5[\/latex]\r\n\r\nn<\/em> = \u22121\r\n\r\n[latex]p - 11=-4[\/latex]\r\n\r\n[latex]-9r=-54[\/latex]\r\n\r\nr<\/em> = 6\r\n\r\nIn the following exercises, translate and solve.\r\n\r\n[latex]\\text{The product of 15 and}x\\text{is 75.}[\/latex]\r\n\r\n[latex]\\text{Eight less than}y\\text{is -32.}[\/latex]\r\n\r\ny<\/em> \u2212 8 = \u221232; y<\/em> = \u221224\r\n\r\n<\/div>","rendered":"

                                                                      Classes of Real Numbers<\/h3>\n

                                                                      Rational Numbers<\/strong><\/h4>\n

                                                                      In the following exercises, write as the ratio of two integers.<\/p>\n

                                                                        \n
                                                                      1. \u24d0 [latex]5[\/latex]<\/li>\n
                                                                      2. \u24d1 [latex]3.19[\/latex]<\/li>\n<\/ol>\n
                                                                          \n
                                                                        1. \u24d0 [latex]\\Large\\frac{5}{1}[\/latex]<\/li>\n
                                                                        2. \u24d1 [latex]\\Large\\frac{319}{100}[\/latex]<\/li>\n<\/ol>\n
                                                                            \n
                                                                          1. \u24d0 [latex]8[\/latex]<\/li>\n
                                                                          2. \u24d1 [latex]-1.61[\/latex]<\/li>\n<\/ol>\n
                                                                              \n
                                                                            1. \u24d0 [latex]-12[\/latex]<\/li>\n
                                                                            2. \u24d1 [latex]9.279[\/latex]<\/li>\n<\/ol>\n
                                                                                \n
                                                                              1. \u24d0 [latex]\\Large\\frac{-12}{1}[\/latex]<\/li>\n
                                                                              2. \u24d1 [latex]\\Large\\frac{9279}{1000}[\/latex]<\/li>\n<\/ol>\n
                                                                                  \n
                                                                                1. \u24d0 [latex]-16[\/latex]<\/li>\n
                                                                                2. \u24d1 [latex]4.399[\/latex]<\/li>\n<\/ol>\n

                                                                                  In the following exercises, determine which of the given numbers are rational and which are irrational.<\/p>\n

                                                                                  [latex]0.75[\/latex] , [latex]0.22\\stackrel{\\text{-}}{\\text{3}}[\/latex] , [latex]\\text{1.39174... }[\/latex]<\/p>\n

                                                                                  Rational: [latex]0.75,0.22\\stackrel{\\text{-}}{\\text{3}}[\/latex] . Irrational: [latex]\\text{1.39174... }[\/latex]<\/p>\n

                                                                                  [latex]0.36[\/latex] , [latex]\\text{0.94729\\ldots }[\/latex] , [latex]2.52\\stackrel{\\text{-}}{\\text{8}}[\/latex]<\/p>\n

                                                                                  [latex]0.\\stackrel{\\text{-}}{45}[\/latex] , [latex]\\text{1.919293... }[\/latex] , [latex]3.59[\/latex]<\/p>\n

                                                                                  Rational: [latex]0.\\stackrel{\\text{-}}{45}[\/latex] , [latex]3.59[\/latex] . Irrational: [latex]\\text{1.919293... }[\/latex]<\/p>\n

                                                                                  [latex]0.1\\stackrel{\\text{-}}{\\text{3}},\\text{0.42982... }[\/latex] , [latex]1.875[\/latex]<\/p>\n

                                                                                  In the following exercises, identify whether each number is rational or irrational.<\/p>\n

                                                                                    \n
                                                                                  1. \u24d0 [latex]\\sqrt{25}[\/latex]<\/li>\n
                                                                                  2. \u24d1 [latex]\\sqrt{30}[\/latex]<\/li>\n<\/ol>\n
                                                                                      \n
                                                                                    1. \u24d0 rational<\/li>\n
                                                                                    2. \u24d1 irrational<\/li>\n<\/ol>\n
                                                                                        \n
                                                                                      1. \u24d0 [latex]\\sqrt{44}[\/latex]<\/li>\n
                                                                                      2. \u24d1 [latex]\\sqrt{49}[\/latex]<\/li>\n<\/ol>\n
                                                                                          \n
                                                                                        1. \u24d0 [latex]\\sqrt{164}[\/latex]<\/li>\n
                                                                                        2. \u24d1 [latex]\\sqrt{169}[\/latex]<\/li>\n<\/ol>\n
                                                                                            \n
                                                                                          1. \u24d0 irrational<\/li>\n
                                                                                          2. \u24d1 rational<\/li>\n<\/ol>\n
                                                                                              \n
                                                                                            1. \u24d0 [latex]\\sqrt{225}[\/latex]<\/li>\n
                                                                                            2. \u24d1 [latex]\\sqrt{216}[\/latex]<\/li>\n<\/ol>\n

                                                                                              Classifying Real Numbers<\/strong><\/h3>\n

                                                                                              In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.<\/p>\n

                                                                                              [latex]-8[\/latex] , [latex]0,\\text{1.95286....}[\/latex] , [latex]\\Large\\frac{12}{5}[\/latex] , [latex]\\sqrt{36}[\/latex] , [latex]9[\/latex]<\/p>\n

                                                                                              \"The<\/p>\n

                                                                                              [latex]-9[\/latex] , [latex]-3\\Large\\frac{4}{9}[\/latex] , [latex]-\\sqrt{9}[\/latex] , [latex]0.4\\stackrel{\\text{-}}{09}[\/latex] , [latex]\\Large\\frac{11}{6}[\/latex] , [latex]7[\/latex]<\/p>\n

                                                                                              [latex]-\\sqrt{100}[\/latex] , [latex]-7[\/latex] , [latex]-\\Large\\frac{8}{3}[\/latex] , [latex]-1[\/latex] , [latex]0.77[\/latex] , [latex]3\\Large\\frac{1}{4}[\/latex]<\/p>\n

                                                                                              \"The<\/p>\n

                                                                                              \n

                                                                                              EVERYDAY MATH<\/h3>\n

                                                                                              Field trip<\/strong><\/h4>\n

                                                                                              All the [latex]5\\text{th}[\/latex] graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be [latex]147[\/latex] people. Each bus holds [latex]44[\/latex] people.
                                                                                              \n\u24d0 How many buses will be needed?
                                                                                              \n\u24d1 Why must the answer be a whole number?
                                                                                              \n\u24d2 Why shouldn’t you round the answer the usual way?<\/p>\n

                                                                                              Child care<\/strong><\/h4>\n

                                                                                              Serena wants to open a licensed child care center. Her state requires that there be no more than [latex]12[\/latex] children for each teacher. She would like her child care center to serve [latex]40[\/latex] children.
                                                                                              \n\u24d0 How many teachers will be needed?
                                                                                              \n\u24d1 Why must the answer be a whole number?
                                                                                              \n\u24d2 Why shouldn’t you round the answer the usual way?<\/p>\n

                                                                                              \u24d0 4
                                                                                              \n\u24d1 Teachers cannot be divided
                                                                                              \n\u24d2 It would result in a lower number.<\/p>\n

                                                                                              <\/h2>\n<\/div>\n

                                                                                               <\/p>\n

                                                                                              \n

                                                                                              WRITING EXERCISES<\/h3>\n

                                                                                              In your own words, explain the difference between a rational number and an irrational number.<\/p>\n

                                                                                              Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.<\/p>\n

                                                                                              Answers will vary.<\/p>\n<\/div>\n

                                                                                               <\/p>\n

                                                                                              <\/h2>\n

                                                                                              Using the Commutative and Associative Properties<\/h3>\n

                                                                                              In the following exercises, use the commutative properties to rewrite the given expression.<\/p>\n

                                                                                              [latex]\\text{8+9=___}[\/latex]<\/p>\n

                                                                                              [latex]\\text{7+6=___}[\/latex]<\/p>\n

                                                                                              7 + 6 = 6 + 7<\/p>\n

                                                                                              [latex]\\text{8}\\left(-12\\right)\\text{=___}[\/latex]<\/p>\n

                                                                                              [latex]\\text{7}\\left(-13\\right)\\text{=___}[\/latex]<\/p>\n

                                                                                              7(\u221213) = (\u221213)7<\/p>\n

                                                                                              [latex]\\text{(-19)(-14)=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(-12\\right)\\left(-18\\right)\\text{=___}[\/latex]<\/p>\n

                                                                                              (\u221212)(\u221218) = (\u221218)(\u221212)<\/p>\n

                                                                                              [latex]\\text{-11+8=___}[\/latex]<\/p>\n

                                                                                              [latex]\\text{-15+7=___}[\/latex]<\/p>\n

                                                                                              \u221215 + 7 = 7 + (\u221215)<\/p>\n

                                                                                              [latex]\\text{x+4=___}[\/latex]<\/p>\n

                                                                                              [latex]\\text{y+1=___}[\/latex]<\/p>\n

                                                                                              y<\/em> + 1 = 1 + y<\/em><\/p>\n

                                                                                              [latex]\\text{-2a=___}[\/latex]<\/p>\n

                                                                                              [latex]\\text{-3m=___}[\/latex]<\/p>\n

                                                                                              \u22123m<\/em> = m<\/em>(\u22123)<\/p>\n

                                                                                              In the following exercises, use the associative properties to rewrite the given expression.<\/p>\n

                                                                                              [latex]\\left(11+9\\right)\\text{+14=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(21+14\\right)\\text{+9=___}[\/latex]<\/p>\n

                                                                                              (21 + 14) + 9 = 21 + (14 + 9)<\/p>\n

                                                                                              [latex]\\left(12\\cdot 5\\right)\\cdot \\text{7=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(14\\cdot 6\\right)\\cdot \\text{9=___}[\/latex]<\/p>\n

                                                                                              (14 \u00b7 6) \u00b7 9 = 14(6 \u00b7 9)<\/p>\n

                                                                                              [latex]\\left(-7+9\\right)\\text{+8=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(-2+6\\right)\\text{+7=___}[\/latex]<\/p>\n

                                                                                              (\u22122 + 6) + 7 = \u22122 + (6 + 7)<\/p>\n

                                                                                              [latex]\\left(16\\cdot\\Large\\frac{4}{5}\\normalsize\\right)\\cdot \\text{15=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(13\\cdot\\Large\\frac{2}{3}\\normalsize\\right)\\cdot \\text{18=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(13\\cdot\\Large\\frac{2}{3}\\normalsize\\right)\\cdot 18=13\\left(\\Large\\frac{2}{3}\\normalsize\\cdot 18\\right)[\/latex]<\/p>\n

                                                                                              [latex]3\\left(4x\\right)\\text{=___}[\/latex]<\/p>\n

                                                                                              [latex]4\\left(7x\\right)\\text{=___}[\/latex]<\/p>\n

                                                                                              4(7x<\/em>) = (4 \u00b7 7)x<\/em><\/p>\n

                                                                                              [latex]\\left(12+x\\right)\\text{+28=___}[\/latex]<\/p>\n

                                                                                              [latex]\\left(17+y\\right)\\text{+33=___}[\/latex]<\/p>\n

                                                                                              (17 + y<\/em>) + 33 = 17 + (y<\/em> + 33)<\/p>\n

                                                                                              Evaluate Expressions using the Commutative and Associative Properties<\/strong><\/h3>\n

                                                                                              In the following exercises, evaluate each expression for the given value.<\/p>\n

                                                                                              If [latex]y=\\Large\\frac{5}{8}[\/latex], evaluate:<\/p>\n

                                                                                              \u24d0 [latex]y+0.49+\\left(-y\\right)[\/latex]
                                                                                              \n\u24d1 [latex]y+\\left(-y\\right)+0.49[\/latex]<\/p>\n

                                                                                              If [latex]z=\\Large\\frac{7}{8}[\/latex], evaluate:<\/p>\n

                                                                                              \u24d0 [latex]z+0.97+\\left(-z\\right)[\/latex]
                                                                                              \n\u24d1 [latex]z+\\left(-z\\right)+0.97[\/latex]<\/p>\n

                                                                                                \n
                                                                                              1. \u24d0 0.97<\/li>\n
                                                                                              2. \u24d1 0.97<\/li>\n<\/ol>\n

                                                                                                If [latex]c=-\\Large\\frac{11}{4}[\/latex], evaluate:<\/p>\n

                                                                                                \u24d0 [latex]c+3.125+\\left(-c\\right)[\/latex]
                                                                                                \n\u24d1 [latex]c+\\left(-c\\right)+3.125[\/latex]<\/p>\n

                                                                                                If [latex]d=-\\Large\\frac{9}{4}[\/latex], evaluate:<\/p>\n

                                                                                                \u24d0 [latex]d+2.375+\\left(-d\\right)[\/latex]
                                                                                                \n\u24d1 [latex]d+\\left(-d\\right)+2.375[\/latex]<\/p>\n

                                                                                                \u24d0 2.375
                                                                                                \n\u24d1 2.375<\/p>\n

                                                                                                If [latex]j=11[\/latex], evaluate:<\/p>\n

                                                                                                \u24d0 [latex]\\Large\\frac{5}{6}\\left(\\frac{6}{5}\\normalsize j\\Large\\right)[\/latex]
                                                                                                \n\u24d1 [latex]\\Large\\left(\\frac{5}{6}\\normalsize\\cdot\\Large\\frac{6}{5}\\right)\\normalsize j[\/latex]<\/p>\n

                                                                                                If [latex]k=21[\/latex], evaluate:<\/p>\n

                                                                                                \u24d0 [latex]\\Large\\frac{4}{13}\\left(\\frac{13}{4}\\normalsize k\\Large\\right)[\/latex]
                                                                                                \n\u24d1 [latex]\\Large\\left(\\frac{4}{13}\\normalsize\\cdot\\Large\\frac{13}{4}\\right)\\normalsize k[\/latex]<\/p>\n

                                                                                                \u24d0 21
                                                                                                \n\u24d1 21<\/p>\n

                                                                                                If [latex]m=-25[\/latex], evaluate:<\/p>\n

                                                                                                \u24d0 [latex]-\\Large\\frac{3}{7}\\left(\\frac{7}{3}\\normalsize m\\Large\\right)[\/latex]
                                                                                                \n\u24d1 [latex]\\Large\\left(-\\frac{3}{7}\\normalsize\\cdot\\Large\\frac{7}{3}\\right)\\normalsize m[\/latex]<\/p>\n

                                                                                                If [latex]n=-8[\/latex], evaluate:<\/p>\n

                                                                                                \u24d0 [latex]-\\Large\\frac{5}{21}\\left(\\frac{21}{5}\\normalsize n\\Large\\right)[\/latex]
                                                                                                \n\u24d1 [latex]\\Large\\left(-\\frac{5}{21}\\normalsize\\cdot\\Large\\frac{21}{5}\\right)\\normalsize n[\/latex]<\/p>\n

                                                                                                \u24d0 \u22128
                                                                                                \n\u24d1 \u22128<\/p>\n

                                                                                                Simplify Expressions Using the Commutative and Associative Properties<\/strong><\/h3>\n

                                                                                                In the following exercises, simplify.<\/p>\n

                                                                                                [latex]-45a+15+45a[\/latex]<\/p>\n

                                                                                                [latex]9y+23+\\left(-9y\\right)[\/latex]<\/p>\n

                                                                                                23<\/p>\n

                                                                                                [latex]\\Large\\frac{1}{2}\\normalsize +\\Large\\frac{7}{8}\\normalsize +\\Large\\left(-\\frac{1}{2}\\right)[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{2}{5}\\normalsize +\\Large\\frac{5}{12}\\normalsize+\\Large\\left(-\\frac{2}{5}\\right)[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{5}{12}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{3}{20}\\normalsize\\cdot\\Large\\frac{49}{11}\\normalsize\\cdot\\Large\\frac{20}{3}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{13}{18}\\normalsize\\cdot\\Large\\frac{25}{7}\\normalsize\\cdot\\Large\\frac{18}{13}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{25}{7}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{7}{12}\\normalsize\\cdot\\Large\\frac{9}{17}\\normalsize\\cdot\\Large\\frac{24}{7}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{3}{10}\\normalsize\\cdot\\Large\\frac{13}{23}\\normalsize\\cdot\\Large\\frac{50}{3}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{65}{23}[\/latex]<\/p>\n

                                                                                                [latex]-24\\cdot 7\\cdot\\Large\\frac{3}{8}[\/latex]<\/p>\n

                                                                                                [latex]-36\\cdot 11\\cdot\\Large\\frac{4}{9}[\/latex]<\/p>\n

                                                                                                \u2212176<\/p>\n

                                                                                                [latex]\\Large\\left(\\frac{5}{6}\\normalsize +\\Large\\frac{8}{15}\\right)\\normalsize +\\Large\\frac{7}{15}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\left(\\frac{1}{12}\\normalsize +\\Large\\frac{4}{9}\\right)\\normalsize +\\Large\\frac{5}{9}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{13}{12}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{5}{13}\\normalsize +\\Large\\frac{3}{4}\\normalsize +\\Large\\frac{1}{4}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{8}{15}\\normalsize +\\Large\\frac{5}{7}\\normalsize +\\Large\\frac{2}{7}[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{23}{15}[\/latex]<\/p>\n

                                                                                                [latex]\\left(4.33p+1.09p\\right)+3.91p[\/latex]<\/p>\n

                                                                                                [latex]\\left(5.89d+2.75d\\right)+1.25d[\/latex]<\/p>\n

                                                                                                9.89d<\/em><\/p>\n

                                                                                                [latex]17\\left(0.25\\right)\\left(4\\right)[\/latex]<\/p>\n

                                                                                                [latex]36\\left(0.2\\right)\\left(5\\right)[\/latex]<\/p>\n

                                                                                                36<\/p>\n

                                                                                                [latex]\\left[2.48\\left(12\\right)\\right]\\left(0.5\\right)[\/latex]<\/p>\n

                                                                                                [latex]\\left[9.731\\left(4\\right)\\right]\\left(0.75\\right)[\/latex]<\/p>\n

                                                                                                29.193<\/p>\n

                                                                                                [latex]7\\left(4a\\right)[\/latex]<\/p>\n

                                                                                                [latex]9\\left(8w\\right)[\/latex]<\/p>\n

                                                                                                72w<\/em><\/p>\n

                                                                                                [latex]-15\\left(5m\\right)[\/latex]<\/p>\n

                                                                                                [latex]-23\\left(2n\\right)[\/latex]<\/p>\n

                                                                                                \u221246n<\/em><\/p>\n

                                                                                                [latex]12\\left(\\Large\\frac{5}{6}\\normalsize p\\right)[\/latex]<\/p>\n

                                                                                                [latex]20\\left(\\Large\\frac{3}{5}\\normalsize q\\right)[\/latex]<\/p>\n

                                                                                                12q<\/em><\/p>\n

                                                                                                [latex]14x+19y+25x+3y[\/latex]<\/p>\n

                                                                                                [latex]15u+11v+27u+19v[\/latex]<\/p>\n

                                                                                                42u<\/em> + 30v<\/em><\/p>\n

                                                                                                [latex]43m+\\left(-12n\\right)+\\text{ }\\left(-16m\\right)+\\left(-9n\\right)[\/latex]<\/p>\n

                                                                                                [latex]-22p+17q+\\left(-35p\\right)+\\left(-27q\\right)[\/latex]<\/p>\n

                                                                                                \u221257p<\/em> + (\u221210q<\/em>)<\/p>\n

                                                                                                [latex]\\Large\\frac{3}{8}\\normalsize g+\\Large\\frac{1}{12}\\normalsize h+\\Large\\frac{7}{8}\\normalsize g+\\Large\\frac{5}{12}\\normalsize h[\/latex]<\/p>\n

                                                                                                [latex]\\Large\\frac{5}{6}\\normalsize a+\\Large\\frac{3}{10}\\normalsize b+\\Large\\frac{1}{6}\\normalsize a+\\Large\\frac{9}{10}\\normalsize b[\/latex]<\/p>\n

                                                                                                [latex]a+\\Large\\frac{6}{5}\\normalsize b[\/latex]<\/p>\n

                                                                                                [latex]6.8p+9.14q+\\left(-4.37p\\right)+\\left(-0.88q\\right)[\/latex]<\/p>\n

                                                                                                [latex]9.6m+7.22n+\\left(-2.19m\\right)+\\left(-0.65n\\right)[\/latex]<\/p>\n

                                                                                                7.41m<\/em> + 6.57n<\/em><\/p>\n

                                                                                                \n

                                                                                                EVERYDAY MATH<\/h3>\n

                                                                                                Stamps<\/strong><\/h4>\n

                                                                                                Allie and Loren need to buy stamps. Allie needs four [latex]{$0.49}[\/latex] stamps and nine [latex]{$0.02}[\/latex] stamps. Loren needs eight [latex]{$0.49}[\/latex] stamps and three [latex]{$0.02}[\/latex] stamps.
                                                                                                \n\u24d0 How much will Allie\u2019s stamps cost?
                                                                                                \n\u24d1 How much will Loren\u2019s stamps cost?
                                                                                                \n\u24d2 What is the total cost of the girls\u2019 stamps?
                                                                                                \n\u24d3 How many [latex]{$0.49}[\/latex] stamps do the girls need altogether? How much will they cost?
                                                                                                \n\u24d4 How many [latex]{$0.02}[\/latex] stamps do the girls need altogether? How much will they cost?<\/p>\n

                                                                                                Counting Cash<\/strong><\/h4>\n

                                                                                                Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three [latex]{$5}[\/latex] bills, eighteen [latex]{$10}[\/latex] bills, and thirty-four [latex]{$20}[\/latex] bills. In another envelope, he has fourteen [latex]{$5}[\/latex] bills, nine [latex]{$10}[\/latex] bills, and twenty-seven [latex]{$20}[\/latex] bills.
                                                                                                \n\u24d0 How much money is in the first envelope?
                                                                                                \n\u24d1 How much money is in the second envelope?
                                                                                                \n\u24d2 What is the total value of all the cash?
                                                                                                \n\u24d3 What is the value of all the [latex]{$5}[\/latex] bills?
                                                                                                \n\u24d4 What is the value of all [latex]{$10}[\/latex] bills?
                                                                                                \n\u24d5 What is the value of all [latex]{$20}[\/latex] bills?<\/p>\n

                                                                                                  \n
                                                                                                1. \u24d0 $975<\/li>\n
                                                                                                2. \u24d1 $700<\/li>\n
                                                                                                3. \u24d2 $1675<\/li>\n
                                                                                                4. \u24d3 $185<\/li>\n
                                                                                                5. \u24d4 $270<\/li>\n
                                                                                                6. \u24d5 $1220<\/li>\n<\/ol>\n<\/div>\n

                                                                                                   <\/p>\n

                                                                                                  \n

                                                                                                  WRITING EXERCISES<\/h3>\n

                                                                                                  In your own words, state the Commutative Property of Addition and explain why it is useful.<\/p>\n

                                                                                                  In your own words, state the Associative Property of Multiplication and explain why it is useful.<\/p>\n

                                                                                                  Answers will vary.<\/p>\n

                                                                                                  <\/h2>\n<\/div>\n

                                                                                                   <\/p>\n

                                                                                                  Using the Distributive Property<\/h2>\n

                                                                                                  Simplify Expressions Using the Distributive Property<\/strong><\/h3>\n

                                                                                                  In the following exercises, simplify using the distributive property.<\/p>\n

                                                                                                  [latex]4\\left(x+8\\right)[\/latex]<\/p>\n

                                                                                                  [latex]3\\left(a+9\\right)[\/latex]<\/p>\n

                                                                                                  3a<\/em> + 27<\/p>\n

                                                                                                  [latex]8\\left(4y+9\\right)[\/latex]<\/p>\n

                                                                                                  [latex]9\\left(3w+7\\right)[\/latex]<\/p>\n

                                                                                                  27w<\/em> + 63<\/p>\n

                                                                                                  [latex]6\\left(c - 13\\right)[\/latex]<\/p>\n

                                                                                                  [latex]7\\left(y - 13\\right)[\/latex]<\/p>\n

                                                                                                  7y<\/em> \u2212 91<\/p>\n

                                                                                                  [latex]7\\left(3p - 8\\right)[\/latex]<\/p>\n

                                                                                                  [latex]5\\left(7u - 4\\right)[\/latex]<\/p>\n

                                                                                                  35u<\/em> \u2212 20<\/p>\n

                                                                                                  [latex]\\Large\\frac{1}{2}\\normalsize\\left(n+8\\right)[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{1}{3}\\normalsize\\left(u+9\\right)[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{1}{3}\\normalsize u+3[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{1}{4}\\normalsize\\left(3q+12\\right)[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{1}{5}\\normalsize\\left(4m+20\\right)[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{4}{5}\\normalsize m+4[\/latex]<\/p>\n

                                                                                                  [latex]9\\left(\\Large\\frac{5}{9}\\normalsize y-\\Large\\frac{1}{3}\\normalsize\\right)[\/latex]<\/p>\n

                                                                                                  [latex]10\\left(\\Large\\frac{3}{10}\\normalsize x-\\Large\\frac{2}{5}\\normalsize\\right)[\/latex]<\/p>\n

                                                                                                  3x<\/em> \u2212 4<\/p>\n

                                                                                                  [latex]12\\left(\\Large\\frac{1}{4}\\normalsize+\\Large\\frac{2}{3}\\normalsize r\\right)[\/latex]<\/p>\n

                                                                                                  [latex]12\\left(\\Large\\frac{1}{6}\\normalsize +\\Large\\frac{3}{4}\\normalsize s\\right)[\/latex]<\/p>\n

                                                                                                  2 + 9s<\/em><\/p>\n

                                                                                                  [latex]r\\left(s - 18\\right)[\/latex]<\/p>\n

                                                                                                  [latex]u\\left(v - 10\\right)[\/latex]<\/p>\n

                                                                                                  uv<\/em> \u2212 10u<\/em><\/p>\n

                                                                                                  [latex]\\left(y+4\\right)p[\/latex]<\/p>\n

                                                                                                  [latex]\\left(a+7\\right)x[\/latex]<\/p>\n

                                                                                                  ax<\/em> + 7x<\/em><\/p>\n

                                                                                                  [latex]-2\\left(y+13\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-3\\left(a+11\\right)[\/latex]<\/p>\n

                                                                                                  \u22123a<\/em> \u2212 33<\/p>\n

                                                                                                  [latex]-7\\left(4p+1\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-9\\left(9a+4\\right)[\/latex]<\/p>\n

                                                                                                  \u221281a<\/em> \u2212 36<\/p>\n

                                                                                                  [latex]-3\\left(x - 6\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-4\\left(q - 7\\right)[\/latex]<\/p>\n

                                                                                                  \u22124q<\/em> + 28<\/p>\n

                                                                                                  [latex]-9\\left(3a - 7\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-6\\left(7x - 8\\right)[\/latex]<\/p>\n

                                                                                                  \u221242x<\/em> + 48<\/p>\n

                                                                                                  [latex]-\\left(r+7\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-\\left(q+11\\right)[\/latex]<\/p>\n

                                                                                                  \u2212q<\/em> \u2212 11<\/p>\n

                                                                                                  [latex]-\\left(3x - 7\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-\\left(5p - 4\\right)[\/latex]<\/p>\n

                                                                                                  \u22125p<\/em> + 4<\/p>\n

                                                                                                  [latex]5+9\\left(n - 6\\right)[\/latex]<\/p>\n

                                                                                                  [latex]12+8\\left(u - 1\\right)[\/latex]<\/p>\n

                                                                                                  8u<\/em> + 4<\/p>\n

                                                                                                  [latex]16 - 3\\left(y+8\\right)[\/latex]<\/p>\n

                                                                                                  [latex]18 - 4\\left(x+2\\right)[\/latex]<\/p>\n

                                                                                                  \u22124x<\/em> + 10<\/p>\n

                                                                                                  [latex]4 - 11\\left(3c - 2\\right)[\/latex]<\/p>\n

                                                                                                  [latex]9 - 6\\left(7n - 5\\right)[\/latex]<\/p>\n

                                                                                                  \u221242n<\/em> + 39<\/p>\n

                                                                                                  [latex]22-\\left(a+3\\right)[\/latex]<\/p>\n

                                                                                                  [latex]8-\\left(r - 7\\right)[\/latex]<\/p>\n

                                                                                                  \u2212r<\/em> + 15<\/p>\n

                                                                                                  [latex]-12-\\left(u+10\\right)[\/latex]<\/p>\n

                                                                                                  [latex]-4-\\left(c - 10\\right)[\/latex]<\/p>\n

                                                                                                  \u2212c<\/em> + 6<\/p>\n

                                                                                                  [latex]\\left(5m - 3\\right)-\\left(m+7\\right)[\/latex]<\/p>\n

                                                                                                  [latex]\\left(4y - 1\\right)-\\left(y - 2\\right)[\/latex]<\/p>\n

                                                                                                  3y<\/em> + 1<\/p>\n

                                                                                                  [latex]5\\left(2n+9\\right)+12\\left(n - 3\\right)[\/latex]<\/p>\n

                                                                                                  [latex]9\\left(5u+8\\right)+2\\left(u - 6\\right)[\/latex]<\/p>\n

                                                                                                  47u<\/em> + 60<\/p>\n

                                                                                                  [latex]9\\left(8x - 3\\right)-\\left(-2\\right)[\/latex]<\/p>\n

                                                                                                  [latex]4\\left(6x - 1\\right)-\\left(-8\\right)[\/latex]<\/p>\n

                                                                                                  24x<\/em> + 4<\/p>\n

                                                                                                  [latex]14\\left(c - 1\\right)-8\\left(c - 6\\right)[\/latex]<\/p>\n

                                                                                                  [latex]11\\left(n - 7\\right)-5\\left(n - 1\\right)[\/latex]<\/p>\n

                                                                                                  6n<\/em> \u2212 72<\/p>\n

                                                                                                  [latex]6\\left(7y+8\\right)-\\left(30y - 15\\right)[\/latex]<\/p>\n

                                                                                                  [latex]7\\left(3n+9\\right)-\\left(4n - 13\\right)[\/latex]<\/p>\n

                                                                                                  17n<\/em> + 76<\/p>\n

                                                                                                  Evaluate Expressions Using the Distributive Property<\/strong><\/h3>\n

                                                                                                  In the following exercises, evaluate both expressions for the given value.<\/p>\n

                                                                                                  If [latex]v=-2[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]6\\left(4v+7\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]6\\cdot 4v+6\\cdot 7[\/latex]<\/p>\n

                                                                                                  If [latex]u=-1[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]8\\left(5u+12\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]8\\cdot 5u+8\\cdot 12[\/latex]<\/p>\n

                                                                                                  \u24d0 56
                                                                                                  \n\u24d1 56<\/p>\n

                                                                                                  If [latex]n=\\Large\\frac{2}{3}[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]3\\left(n+\\Large\\frac{5}{6}\\normalsize\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]3\\cdot n+3\\cdot\\Large\\frac{5}{6}[\/latex]<\/p>\n

                                                                                                  If [latex]y=\\Large\\frac{3}{4}[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]4\\left(y+\\Large\\frac{3}{8}\\normalsize\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]4\\cdot y+4\\cdot\\Large\\frac{3}{8}[\/latex]<\/p>\n

                                                                                                  \u24d0 [latex]\\Large\\frac{9}{2}[\/latex]
                                                                                                  \n\u24d1 [latex]\\Large\\frac{9}{2}[\/latex]<\/p>\n

                                                                                                  If [latex]y=\\Large\\frac{7}{12}[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-3\\left(4y+15\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]3\\cdot 4y+\\left(-3\\right)\\cdot 15[\/latex]<\/p>\n

                                                                                                  If [latex]p=\\Large\\frac{23}{30}[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-6\\left(5p+11\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-6\\cdot 5p+\\left(-6\\right)\\cdot 11[\/latex]<\/p>\n

                                                                                                  \u24d0 \u221289
                                                                                                  \n\u24d1 \u221289<\/p>\n

                                                                                                  If [latex]m=0.4[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-10\\left(3m - 0.9\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-10\\cdot 3m-\\left(-10\\right)\\left(0.9\\right)[\/latex]<\/p>\n

                                                                                                  If [latex]n=0.75[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-100\\left(5n+1.5\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-100\\cdot 5n+\\left(-100\\right)\\left(1.5\\right)[\/latex]<\/p>\n

                                                                                                  \u24d0 \u2212525
                                                                                                  \n\u24d1 \u2212525<\/p>\n

                                                                                                  If [latex]y=-25[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-\\left(y - 25\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-y+25[\/latex]<\/p>\n

                                                                                                  If [latex]w=-80[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-\\left(w - 80\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-w+80[\/latex]<\/p>\n

                                                                                                  \u24d0 160
                                                                                                  \n\u24d1 160<\/p>\n

                                                                                                  If [latex]p=0.19[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-\\left(p+0.72\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-p - 0.72[\/latex]<\/p>\n

                                                                                                  If [latex]q=0.55[\/latex], evaluate<\/p>\n

                                                                                                  \u24d0 [latex]-\\left(q+0.48\\right)[\/latex]
                                                                                                  \n\u24d1 [latex]-q - 0.48[\/latex]<\/p>\n

                                                                                                  \u24d0 \u22121.03
                                                                                                  \n\u24d1 \u22121.03<\/p>\n

                                                                                                  \n

                                                                                                  EVERYDAY MATH<\/h3>\n

                                                                                                  Buying by the case<\/strong><\/h4>\n

                                                                                                  Joe can buy his favorite ice tea at a convenience store for [latex]{$1.99}[\/latex] per bottle. At the grocery store, he can buy a case of [latex]12[\/latex] bottles for [latex]{$23.88}[\/latex].
                                                                                                  \n\u24d0 Use the distributive property to find the cost of [latex]12[\/latex] bottles bought individually at the convenience store. (Hint: notice that [latex]{$1.99}[\/latex] is [latex]{$2}-{$0.01}[\/latex]. )
                                                                                                  \n\u24d1 Is it a bargain to buy the iced tea at the grocery store by the case?<\/p>\n

                                                                                                  Multi-pack purchase<\/strong><\/h4>\n

                                                                                                  Adele\u2019s shampoo sells for [latex]{$3.97}[\/latex] per bottle at the drug store. At the warehouse store, the same shampoo is sold as a [latex]{3-pack}[\/latex] for [latex]{$10.49}[\/latex].
                                                                                                  \n\u24d0 Show how you can use the distributive property to find the cost of [latex]3[\/latex] bottles bought individually at the drug store.
                                                                                                  \n\u24d1 How much would Adele save by buying the [latex]{3-pack}[\/latex] at the warehouse store?<\/p>\n

                                                                                                  \u24d0 3(4 \u2212 0.03) = 11.91
                                                                                                  \n\u24d1 $1.42<\/p>\n

                                                                                                  <\/h2>\n<\/div>\n

                                                                                                   <\/p>\n

                                                                                                  \n

                                                                                                  WRITING EXERCISES<\/h3>\n

                                                                                                  Simplify [latex]8\\left(x-\\Large\\frac{1}{4}\\normalsize\\right)[\/latex] using the distributive property and explain each step.<\/p>\n

                                                                                                  Explain how you can multiply [latex]4\\left({$5.97}\\right)[\/latex] without paper or a calculator by thinking of [latex]{$5.97}[\/latex] as [latex]6 - 0.03[\/latex] and then using the distributive property.<\/p>\n<\/div>\n

                                                                                                   <\/p>\n

                                                                                                  <\/h2>\n

                                                                                                  Properties of Identity, Inverses, and Zero<\/h2>\n

                                                                                                  In the following exercises, identify whether each example is using the identity property of addition or multiplication.<\/p>\n

                                                                                                  [latex]101+0=101[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{3}{5}\\normalsize\\left(1\\right)=\\Large\\frac{3}{5}[\/latex]<\/p>\n

                                                                                                  identity property of multiplication<\/p>\n

                                                                                                  [latex]-9\\cdot 1=-9[\/latex]<\/p>\n

                                                                                                  [latex]0+64=64[\/latex]<\/p>\n

                                                                                                  identity property of addition<\/p>\n

                                                                                                  Use the Inverse Properties of Addition and Multiplication<\/strong>
                                                                                                  \nIn the following exercises, find the multiplicative inverse.<\/p>\n

                                                                                                  [latex]8[\/latex]<\/p>\n

                                                                                                  [latex]14[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{1}{14}[\/latex]<\/p>\n

                                                                                                  [latex]-17[\/latex]<\/p>\n

                                                                                                  [latex]-19[\/latex]<\/p>\n

                                                                                                  [latex]-\\Large\\frac{1}{19}[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{7}{12}[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{8}{13}[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{13}{8}[\/latex]<\/p>\n

                                                                                                  [latex]-\\Large\\frac{3}{10}[\/latex]<\/p>\n

                                                                                                  [latex]-\\Large\\frac{5}{12}[\/latex]<\/p>\n

                                                                                                  [latex]-\\Large\\frac{12}{5}[\/latex]<\/p>\n

                                                                                                  [latex]0.8[\/latex]<\/p>\n

                                                                                                  [latex]0.4[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{5}{2}[\/latex]<\/p>\n

                                                                                                  [latex]-0.2[\/latex]<\/p>\n

                                                                                                  [latex]-0.5[\/latex]<\/p>\n

                                                                                                  \u22122<\/p>\n

                                                                                                  Use the Properties of Zero<\/strong><\/h3>\n

                                                                                                  In the following exercises, simplify using the properties of zero.<\/p>\n

                                                                                                  [latex]48\\cdot 0[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{0}{6}[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]\\Large\\frac{3}{0}[\/latex]<\/p>\n

                                                                                                  [latex]22\\cdot 0[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]0\\div\\Large\\frac{11}{12}[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{6}{0}[\/latex]<\/p>\n

                                                                                                  undefined<\/p>\n

                                                                                                  [latex]\\Large\\frac{0}{3}[\/latex]<\/p>\n

                                                                                                  [latex]0\\div\\Large\\frac{7}{15}[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]0\\cdot\\Large\\frac{8}{15}[\/latex]<\/p>\n

                                                                                                  [latex]\\left(-3.14\\right)\\left(0\\right)[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]5.72\\div 0[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{\\frac{1}{10}}{0}[\/latex]<\/p>\n

                                                                                                  undefined<\/p>\n

                                                                                                  Simplify Expressions using the Properties of Identities, Inverses, and Zero<\/strong><\/h3>\n

                                                                                                  In the following exercises, simplify using the properties of identities, inverses, and zero.<\/p>\n

                                                                                                  [latex]19a+44 - 19a[\/latex]<\/p>\n

                                                                                                  [latex]27c+16 - 27c[\/latex]<\/p>\n

                                                                                                  16<\/p>\n

                                                                                                  [latex]38+11r - 38[\/latex]<\/p>\n

                                                                                                  [latex]92+31s - 92[\/latex]<\/p>\n

                                                                                                  31s<\/em><\/p>\n

                                                                                                  [latex]10\\left(0.1d\\right)[\/latex]<\/p>\n

                                                                                                  [latex]100\\left(0.01p\\right)[\/latex]<\/p>\n

                                                                                                  p<\/em><\/p>\n

                                                                                                  [latex]5\\left(0.6q\\right)[\/latex]<\/p>\n

                                                                                                  [latex]40\\left(0.05n\\right)[\/latex]<\/p>\n

                                                                                                  2n<\/em><\/p>\n

                                                                                                  [latex]\\Large\\frac{0}{r+20}[\/latex] , where [latex]r\\ne -20[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{0}{s+13}[\/latex] , where [latex]s\\ne -13[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]\\Large\\frac{0}{u - 4.99}[\/latex] , where [latex]u\\ne 4.99[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{0}{v - 65.1}[\/latex] , where [latex]v\\ne 65.1[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]0\\div \\left(x-\\Large\\frac{1}{2}\\normalsize\\right)[\/latex] , where [latex]x\\ne\\Large\\frac{1}{2}[\/latex]<\/p>\n

                                                                                                  [latex]0\\div \\left(y-\\Large\\frac{1}{6}\\normalsize\\right)[\/latex] , where [latex]y\\ne\\Large\\frac{1}{6}[\/latex]<\/p>\n

                                                                                                  0<\/p>\n

                                                                                                  [latex]\\Large\\frac{32 - 5a}{0}[\/latex] , where [latex]32 - 5a\\ne 0[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{28 - 9b}{0}[\/latex] , where [latex]28 - 9b\\ne 0[\/latex]<\/p>\n

                                                                                                  undefined<\/p>\n

                                                                                                  [latex]\\Large\\frac{2.1+0.4c}{0}[\/latex] , where [latex]2.1+0.4c\\ne 0[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{1.75+9f}{0}[\/latex] , where [latex]1.75+9f\\ne 0[\/latex]<\/p>\n

                                                                                                  undefined<\/p>\n

                                                                                                  [latex]\\left(\\Large\\frac{3}{4}\\normalsize +\\Large\\frac{9}{10}\\normalsize m\\right)\\div 0[\/latex] , where [latex]\\Large\\frac{3}{4}\\normalsize +\\Large\\frac{9}{10}\\normalsize m\\ne 0[\/latex]<\/p>\n

                                                                                                  [latex]\\left(\\Large\\frac{5}{16}\\normalsize n-\\Large\\frac{3}{7}\\normalsize\\right)\\div 0[\/latex] , where [latex]\\Large\\frac{5}{16}\\normalsize n-\\Large\\frac{3}{7}\\normalsize \\ne 0[\/latex]<\/p>\n

                                                                                                  undefined<\/p>\n

                                                                                                  [latex]\\Large\\frac{9}{10}\\normalsize\\cdot\\Large\\frac{10}{9}\\normalsize\\left(18p - 21\\right)[\/latex]<\/p>\n

                                                                                                  [latex]\\Large\\frac{5}{7}\\normalsize\\cdot\\Large\\frac{7}{5}\\normalsize\\left(20q - 35\\right)[\/latex]<\/p>\n

                                                                                                  20q<\/em> \u2212 35<\/p>\n

                                                                                                  [latex]15\\cdot\\Large\\frac{3}{5}\\normalsize\\left(4d+10\\right)[\/latex]<\/p>\n

                                                                                                  [latex]18\\cdot\\Large\\frac{5}{6}\\normalsize\\left(15h+24\\right)[\/latex]<\/p>\n

                                                                                                  225h<\/em> + 360<\/p>\n

                                                                                                  \n

                                                                                                  EVERYDAY MATH<\/h3>\n

                                                                                                  Insurance copayment<\/strong><\/h4>\n

                                                                                                  Carrie had to have [latex]5[\/latex] fillings done. Each filling cost [latex]{$80}[\/latex]. Her dental insurance required her to pay [latex]20%[\/latex] of the cost. Calculate Carrie\u2019s cost
                                                                                                  \n\u24d0 by finding her copay for each filling, then finding her total cost for [latex]5[\/latex] fillings, and
                                                                                                  \n\u24d1 by multiplying [latex]5\\left(0.20\\right)\\left(80\\right)[\/latex].
                                                                                                  \n\u24d2 Which of the Properties of Real Numbers did you use for part (b)?<\/p>\n

                                                                                                  Cooking time<\/strong><\/h4>\n

                                                                                                  Helen bought a [latex]\\text{24-pound}[\/latex] turkey for her family\u2019s Thanksgiving dinner and wants to know what time to put the turkey in the oven. She wants to allow [latex]20[\/latex] minutes per pound cooking time.
                                                                                                  \n\u24d0 Calculate the length of time needed to roast the turkey by multiplying [latex]24\\cdot 20[\/latex] to find the number of minutes and then multiplying the product by [latex]\\Large\\frac{1}{60}[\/latex] to convert minutes into hours.
                                                                                                  \n\u24d1 Multiply [latex]24\\left(20\\cdot\\Large\\frac{1}{60}\\normalsize\\right)[\/latex].
                                                                                                  \n\u24d2 Which of the Properties of Real Numbers allows you to multiply [latex]24\\left(20\\cdot\\Large\\frac{1}{60}\\right)[\/latex] instead of [latex]\\left(24\\cdot 20\\right)\\Large\\frac{1}{60}[\/latex]?<\/p>\n

                                                                                                  \u24d0 8 hours
                                                                                                  \n\u24d1 8
                                                                                                  \n\u24d2 associative property of multiplication<\/p>\n

                                                                                                  <\/h2>\n<\/div>\n

                                                                                                   <\/p>\n

                                                                                                  \n

                                                                                                  WRITING EXERCISES<\/h3>\n

                                                                                                  In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.<\/p>\n

                                                                                                  How can the use of the properties of real numbers make it easier to simplify expressions?<\/p>\n

                                                                                                  Answers will vary.<\/p>\n<\/div>\n

                                                                                                   <\/p>\n

                                                                                                  Determine Whether a Number is a Solution of an Equation<\/strong><\/h2>\n

                                                                                                  In the following exercises, determine whether each number is a solution of the given equation.<\/p>\n

                                                                                                  [latex]4x - 2=6[\/latex]<\/p>\n

                                                                                                  \u24d0 [latex]x=-2[\/latex]
                                                                                                  \n\u24d1 [latex]x=-1[\/latex]
                                                                                                  \n\u24d2 [latex]x=2[\/latex]<\/p>\n

                                                                                                    \n
                                                                                                  1. \u24d0 no<\/li>\n
                                                                                                  2. \u24d1 yes<\/li>\n
                                                                                                  3. \u24d2 no<\/li>\n<\/ol>\n

                                                                                                    [latex]4y - 10=-14[\/latex]<\/p>\n

                                                                                                    \u24d0 [latex]y=-6[\/latex]
                                                                                                    \n\u24d1 [latex]y=-1[\/latex]
                                                                                                    \n\u24d2 [latex]y=1[\/latex]<\/p>\n

                                                                                                    [latex]9a+27=-63[\/latex]<\/p>\n

                                                                                                    \u24d0 [latex]a=6[\/latex]
                                                                                                    \n\u24d1 [latex]a=-6[\/latex]
                                                                                                    \n\u24d2 [latex]a=-10[\/latex]<\/p>\n

                                                                                                      \n
                                                                                                    1. \u24d0 no<\/li>\n
                                                                                                    2. \u24d1 no<\/li>\n
                                                                                                    3. \u24d2 yes<\/li>\n<\/ol>\n

                                                                                                      [latex]7c+42=-56[\/latex]<\/p>\n

                                                                                                      \u24d0 [latex]c=2[\/latex]
                                                                                                      \n\u24d1 [latex]c=-2[\/latex]
                                                                                                      \n\u24d2 [latex]c=-14[\/latex]<\/p>\n

                                                                                                      Solve Equations Using the Addition and Subtraction Properties of Equality<\/strong><\/h3>\n

                                                                                                      In the following exercises, solve for the unknown.<\/p>\n

                                                                                                      [latex]n+12=5[\/latex]<\/p>\n

                                                                                                      \u22127<\/p>\n

                                                                                                      [latex]m+16=2[\/latex]<\/p>\n

                                                                                                      [latex]p+9=-8[\/latex]<\/p>\n

                                                                                                      \u221217<\/p>\n

                                                                                                      [latex]q+5=-6[\/latex]<\/p>\n

                                                                                                      [latex]u - 3=-7[\/latex]<\/p>\n

                                                                                                      \u22124<\/p>\n

                                                                                                      [latex]v - 7=-8[\/latex]<\/p>\n

                                                                                                      [latex]h - 10=-4[\/latex]<\/p>\n

                                                                                                      6<\/p>\n

                                                                                                      [latex]k - 9=-5[\/latex]<\/p>\n

                                                                                                      [latex]x+\\left(-2\\right)=-18[\/latex]<\/p>\n

                                                                                                      \u221216<\/p>\n

                                                                                                      [latex]y+\\left(-3\\right)=-10[\/latex]<\/p>\n

                                                                                                      [latex]r-\\left(-5\\right)=-9[\/latex]<\/p>\n

                                                                                                      \u221214<\/p>\n

                                                                                                      [latex]s-\\left(-2\\right)=-11[\/latex]<\/p>\n

                                                                                                      Model the Division Property of Equality<\/strong><\/h3>\n

                                                                                                      In the following exercises, write the equation modeled by the envelopes and counters and then solve it.<\/p>\n

                                                                                                      \"This<\/p>\n

                                                                                                      3x<\/em> = 6; x<\/em> = 2<\/p>\n

                                                                                                      \"This<\/p>\n

                                                                                                      \"This<\/p>\n

                                                                                                      2x<\/em> = 8; x<\/em> = 4<\/p>\n

                                                                                                      \"This<\/p>\n

                                                                                                      Solve Equations Using the Division Property of Equality<\/strong><\/h3>\n

                                                                                                      In the following exercises, solve each equation using the division property of equality and check the solution.<\/p>\n

                                                                                                      [latex]5x=45[\/latex]<\/p>\n

                                                                                                      9<\/p>\n

                                                                                                      [latex]4p=64[\/latex]<\/p>\n

                                                                                                      [latex]-7c=56[\/latex]<\/p>\n

                                                                                                      \u22128<\/p>\n

                                                                                                      [latex]-9x=54[\/latex]<\/p>\n

                                                                                                      [latex]-14p=-42[\/latex]<\/p>\n

                                                                                                      3<\/p>\n

                                                                                                      [latex]-8m=-40[\/latex]<\/p>\n

                                                                                                      [latex]-120=10q[\/latex]<\/p>\n

                                                                                                      \u221212<\/p>\n

                                                                                                      [latex]-75=15y[\/latex]<\/p>\n

                                                                                                      [latex]24x=480[\/latex]<\/p>\n

                                                                                                      20<\/p>\n

                                                                                                      [latex]18n=540[\/latex]<\/p>\n

                                                                                                      [latex]-3z=0[\/latex]<\/p>\n

                                                                                                      0<\/p>\n

                                                                                                      [latex]4u=0[\/latex]<\/p>\n

                                                                                                      Translate to an Equation and Solve<\/strong><\/h3>\n

                                                                                                      In the following exercises, translate and solve.<\/p>\n

                                                                                                      Four more than [latex]n[\/latex] is equal to 1.<\/p>\n

                                                                                                      n<\/em> + 4 = 1; n<\/em> = \u22123<\/p>\n

                                                                                                      Nine more than [latex]m[\/latex] is equal to 5.<\/p>\n

                                                                                                      The sum of eight and [latex]p[\/latex] is [latex]-3[\/latex] .<\/p>\n

                                                                                                      8 + p<\/em> = \u22123; p<\/em> = \u221211<\/p>\n

                                                                                                      The sum of two and [latex]q[\/latex] is [latex]-7[\/latex] .<\/p>\n

                                                                                                      The difference of [latex]a[\/latex] and three is [latex]-14[\/latex] .<\/p>\n

                                                                                                      a<\/em> \u2212 3 = \u221214; a<\/em> = \u221211<\/p>\n

                                                                                                      The difference of [latex]b[\/latex] and [latex]5[\/latex] is [latex]-2[\/latex] .<\/p>\n

                                                                                                      The number \u221242 is the product of \u22127 and [latex]x[\/latex] .<\/p>\n

                                                                                                      \u221242 = \u22127x<\/em>; x<\/em> = 6<\/p>\n

                                                                                                      The number \u221254 is the product of \u22129 and [latex]y[\/latex] .<\/p>\n

                                                                                                      The product of [latex]f[\/latex] and \u221215 is 75.<\/p>\n

                                                                                                      f<\/em>(\u221215) = 75; f<\/em> = 5<\/p>\n

                                                                                                      The product of [latex]g[\/latex] and \u221218 is 36.<\/p>\n

                                                                                                      \u22126 plus [latex]c[\/latex] is equal to 4.<\/p>\n

                                                                                                      \u22126 + c<\/em> = 4; c<\/em> = 10<\/p>\n

                                                                                                      \u22122 plus [latex]d[\/latex] is equal to 1.<\/p>\n

                                                                                                      Nine less than [latex]n[\/latex] is \u22124.<\/p>\n

                                                                                                      m<\/em> \u2212 9 = \u22124; m<\/em> = 5<\/p>\n

                                                                                                      Thirteen less than [latex]n[\/latex] is [latex]-10[\/latex] .<\/p>\n

                                                                                                      Mixed Practice<\/strong><\/h3>\n

                                                                                                      In the following exercises, solve.<\/p>\n

                                                                                                        \n
                                                                                                      1. \u24d0 [latex]x+2=10[\/latex]<\/li>\n
                                                                                                      2. \u24d1 [latex]2x=10[\/latex]<\/li>\n<\/ol>\n
                                                                                                          \n
                                                                                                        1. \u24d0 8<\/li>\n
                                                                                                        2. \u24d1 5<\/li>\n<\/ol>\n
                                                                                                            \n
                                                                                                          1. \u24d0 [latex]y+6=12[\/latex]<\/li>\n
                                                                                                          2. \u24d1 [latex]6y=12[\/latex]<\/li>\n<\/ol>\n
                                                                                                              \n
                                                                                                            1. \u24d0 [latex]-3p=27[\/latex]<\/li>\n
                                                                                                            2. \u24d1 [latex]p - 3=27[\/latex]<\/li>\n<\/ol>\n
                                                                                                                \n
                                                                                                              1. \u24d0 \u22129<\/li>\n
                                                                                                              2. \u24d1 30<\/li>\n<\/ol>\n
                                                                                                                  \n
                                                                                                                1. \u24d0 [latex]-2q=34[\/latex]<\/li>\n
                                                                                                                2. \u24d1 [latex]q - 2=34[\/latex]<\/li>\n<\/ol>\n

                                                                                                                  [latex]a - 4=16[\/latex]<\/p>\n

                                                                                                                  20<\/p>\n

                                                                                                                  [latex]b - 1=11[\/latex]<\/p>\n

                                                                                                                  [latex]-8m=-56[\/latex]<\/p>\n

                                                                                                                  7<\/p>\n

                                                                                                                  [latex]-6n=-48[\/latex]<\/p>\n

                                                                                                                  [latex]-39=u+13[\/latex]<\/p>\n

                                                                                                                  \u221252<\/p>\n

                                                                                                                  [latex]-100=v+25[\/latex]<\/p>\n

                                                                                                                  [latex]11r=-99[\/latex]<\/p>\n

                                                                                                                  \u22129<\/p>\n

                                                                                                                  [latex]15s=-300[\/latex]<\/p>\n

                                                                                                                  [latex]100=20d[\/latex]<\/p>\n

                                                                                                                  5<\/p>\n

                                                                                                                  [latex]250=25n[\/latex]<\/p>\n

                                                                                                                  [latex]-49=x - 7[\/latex]<\/p>\n

                                                                                                                  \u221242<\/p>\n

                                                                                                                  [latex]64=y - 4[\/latex]<\/p>\n

                                                                                                                  \n

                                                                                                                  EVERYDAY MATH<\/h3>\n

                                                                                                                  Cookie packaging<\/strong><\/h4>\n

                                                                                                                  A package of [latex]\\text{51 cookies}[\/latex] has [latex]3[\/latex] equal rows of cookies. Find the number of cookies in each row, [latex]c[\/latex], by solving the equation [latex]3c=51[\/latex].<\/p>\n

                                                                                                                  17 cookies<\/p>\n

                                                                                                                  Kindergarten class<\/strong><\/h4>\n

                                                                                                                  Connie\u2019s kindergarten class has [latex]\\text{24 children.}[\/latex] She wants them to get into [latex]4[\/latex] equal groups. Find the number of children in each group, [latex]g[\/latex], by solving the equation [latex]4g=24[\/latex].<\/p>\n

                                                                                                                  <\/h2>\n<\/div>\n

                                                                                                                   <\/p>\n

                                                                                                                  \n

                                                                                                                  WRITING EXERCISES<\/h3>\n

                                                                                                                  Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation [latex]3x=15?[\/latex] Explain why or why not.<\/p>\n

                                                                                                                  Sample answer: It is helpful because it shows how the counters can be divided among the envelopes.<\/p>\n

                                                                                                                  Suppose you are using envelopes and counters to model solving the equations [latex]x+4=12[\/latex] and [latex]4x=12[\/latex]. Explain how you would solve each equation.<\/p>\n

                                                                                                                  Frida started to solve the equation [latex]-3x=36[\/latex] by adding [latex]3[\/latex] to both sides. Explain why Frida\u2019s method will not solve the equation.<\/p>\n

                                                                                                                  Sample answer: The operation used in the equation is multiplication. The inverse of multiplication is division, not addition.<\/p>\n

                                                                                                                  Raoul started to solve the equation [latex]4y=40[\/latex] by subtracting [latex]4[\/latex] from both sides. Explain why Raoul\u2019s method will not solve the equation.<\/p>\n

                                                                                                                  <\/h2>\n<\/div>\n

                                                                                                                   <\/p>\n

                                                                                                                  Chapter Review Exercises<\/h3>\n
                                                                                                                  \n

                                                                                                                  Introduction to integers<\/h3>\n

                                                                                                                  Locate Positive and Negative Numbers on the Number Line<\/strong><\/h4>\n

                                                                                                                  In the following exercises, locate and label the integer on the number line.<\/p>\n

                                                                                                                  [latex]5[\/latex]<\/p>\n

                                                                                                                  \"This<\/p>\n

                                                                                                                  [latex]-5[\/latex]<\/p>\n

                                                                                                                  [latex]-3[\/latex]<\/p>\n

                                                                                                                  \"This<\/p>\n

                                                                                                                  [latex]3[\/latex]<\/p>\n

                                                                                                                  [latex]-8[\/latex]<\/p>\n

                                                                                                                  \"This<\/p>\n

                                                                                                                  [latex]-7[\/latex]<\/p>\n

                                                                                                                  Order Positive and Negative Numbers<\/strong><\/h4>\n

                                                                                                                  In the following exercises, order each of the following pairs of numbers, using [latex]<[\/latex]; or [latex]\\text{>.}[\/latex]<\/p>\n

                                                                                                                  [latex]4\\text{__}8[\/latex]<\/p>\n

                                                                                                                  <<\/p>\n

                                                                                                                  [latex]-6\\text{__}3[\/latex]<\/p>\n

                                                                                                                  [latex]-5\\text{__}-10[\/latex]<\/p>\n

                                                                                                                  ><\/p>\n

                                                                                                                  [latex]-9\\text{__}-4[\/latex]<\/p>\n

                                                                                                                  [latex]2\\text{__}-7[\/latex]<\/p>\n

                                                                                                                  ><\/p>\n

                                                                                                                  [latex]-3\\text{__}1[\/latex]<\/p>\n

                                                                                                                  Find Opposites<\/strong><\/h4>\n

                                                                                                                  In the following exercises, find the opposite of each number.<\/p>\n

                                                                                                                  [latex]6[\/latex]<\/p>\n

                                                                                                                  \u22126<\/p>\n

                                                                                                                  [latex]-2[\/latex]<\/p>\n

                                                                                                                  [latex]-4[\/latex]<\/p>\n

                                                                                                                  4<\/p>\n

                                                                                                                  [latex]3[\/latex]<\/p>\n

                                                                                                                  In the following exercises, simplify.<\/p>\n

                                                                                                                    \n
                                                                                                                  1. \u24d0 [latex]-\\left(8\\right)[\/latex]<\/li>\n
                                                                                                                  2. \u24d1 [latex]-\\left(-8\\right)[\/latex]<\/li>\n<\/ol>\n
                                                                                                                      \n
                                                                                                                    1. \u24d0 \u22128<\/li>\n
                                                                                                                    2. \u24d1 8<\/li>\n<\/ol>\n
                                                                                                                        \n
                                                                                                                      1. \u24d0 [latex]-\\left(9\\right)[\/latex]<\/li>\n
                                                                                                                      2. \u24d1 [latex]-\\left(-9\\right)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                        In the following exercises, evaluate.<\/p>\n

                                                                                                                        [latex]-x,\\text{when}[\/latex]<\/p>\n

                                                                                                                        \u24d0 [latex]x=32[\/latex]
                                                                                                                        \n\u24d1 [latex]x=-32[\/latex]<\/p>\n

                                                                                                                          \n
                                                                                                                        1. \u24d0 \u221232<\/li>\n
                                                                                                                        2. \u24d1 32<\/li>\n<\/ol>\n

                                                                                                                          [latex]-n,\\text{when}[\/latex]<\/p>\n

                                                                                                                          \u24d0 [latex]n=20[\/latex]
                                                                                                                          \n\u24d0 [latex]n=-20[\/latex]<\/p>\n

                                                                                                                           <\/p>\n<\/div>\n

                                                                                                                           <\/p>\n

                                                                                                                          \n

                                                                                                                          ABSOLUTE VALUES<\/h3>\n

                                                                                                                          Simplify Absolute Values<\/strong><\/h4>\n

                                                                                                                          In the following exercises, simplify.<\/p>\n

                                                                                                                          [latex]|-21|[\/latex]<\/p>\n

                                                                                                                          21<\/p>\n

                                                                                                                          [latex]|-42|[\/latex]<\/p>\n

                                                                                                                          [latex]|36|[\/latex]<\/p>\n

                                                                                                                          36<\/p>\n

                                                                                                                          [latex]-|15|[\/latex]<\/p>\n

                                                                                                                          [latex]|0|[\/latex]<\/p>\n

                                                                                                                          0<\/p>\n

                                                                                                                          [latex]-|-75|[\/latex]<\/p>\n

                                                                                                                          In the following exercises, evaluate.<\/p>\n

                                                                                                                          [latex]|x|\\text{when}x=-14[\/latex]<\/p>\n

                                                                                                                          14<\/p>\n

                                                                                                                          [latex]-|r|\\text{when}r=27[\/latex]<\/p>\n

                                                                                                                          [latex]-|-y|\\text{when}y=33[\/latex]<\/p>\n

                                                                                                                          \u221233<\/p>\n

                                                                                                                          [latex]|{\\text{-n}}|\\text{when}n=-4[\/latex]<\/p>\n

                                                                                                                          In the following exercises, fill in [latex]\\text{<},\\text{>},\\text{or}=[\/latex] for each of the following pairs of numbers.<\/p>\n

                                                                                                                          [latex]-|-4|\\text{__}4[\/latex]<\/p>\n

                                                                                                                          <<\/p>\n

                                                                                                                          [latex]-2\\text{__}|-2|[\/latex]<\/p>\n

                                                                                                                          [latex]-|-6|\\text{__}-6[\/latex]<\/p>\n

                                                                                                                          =<\/p>\n

                                                                                                                          [latex]-|-9|\\text{__}|-9|[\/latex]<\/p>\n

                                                                                                                          In the following exercises, simplify.<\/p>\n

                                                                                                                          [latex]-\\left(-55\\right)\\text{and}-|-55|[\/latex]<\/p>\n

                                                                                                                          \u221255; \u221255<\/p>\n

                                                                                                                          [latex]-\\left(-48\\right)\\text{and}-|-48|[\/latex]<\/p>\n

                                                                                                                          [latex]|12 - 5|[\/latex]<\/p>\n

                                                                                                                          7<\/p>\n

                                                                                                                          [latex]|9+7|[\/latex]<\/p>\n

                                                                                                                          [latex]6|-9|[\/latex]<\/p>\n

                                                                                                                          54<\/p>\n

                                                                                                                          [latex]|14 - 8|-|-2|[\/latex]<\/p>\n

                                                                                                                          [latex]|9 - 3|-|5 - 12|[\/latex]<\/p>\n

                                                                                                                          \u22121<\/p>\n

                                                                                                                          [latex]5+4|15 - 3|[\/latex]<\/p>\n

                                                                                                                           <\/p>\n<\/div>\n

                                                                                                                           <\/p>\n

                                                                                                                          \n

                                                                                                                          Translate Phrases to Expressions with Integers<\/strong><\/h3>\n

                                                                                                                          In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.<\/p>\n

                                                                                                                          the opposite of [latex]16[\/latex]<\/p>\n

                                                                                                                          \u221216<\/p>\n

                                                                                                                          the opposite of [latex]-8[\/latex]<\/p>\n

                                                                                                                          negative [latex]3[\/latex]<\/p>\n

                                                                                                                          \u22123<\/p>\n

                                                                                                                          [latex]19[\/latex] minus negative [latex]12[\/latex]<\/p>\n

                                                                                                                          a temperature of [latex]10[\/latex] below zero<\/p>\n

                                                                                                                          \u221210\u00b0<\/p>\n

                                                                                                                          an elevation of [latex]\\text{85 feet}[\/latex] below sea level<\/p>\n

                                                                                                                          Add Integers<\/p>\n

                                                                                                                          Model Addition of Integers<\/strong><\/h4>\n

                                                                                                                          In the following exercises, model the following to find the sum.<\/p>\n

                                                                                                                          [latex]3+7[\/latex]<\/p>\n

                                                                                                                          10<\/p>\n

                                                                                                                          [latex]-2+6[\/latex]<\/p>\n

                                                                                                                          [latex]5+\\left(-4\\right)[\/latex]<\/p>\n

                                                                                                                          1<\/p>\n

                                                                                                                          [latex]-3+\\left(-6\\right)[\/latex]<\/p>\n

                                                                                                                          Simplify Expressions with Integers<\/strong><\/h4>\n

                                                                                                                          In the following exercises, simplify each expression.<\/p>\n

                                                                                                                          [latex]14+82[\/latex]<\/p>\n

                                                                                                                          96<\/p>\n

                                                                                                                          [latex]-33+\\left(-67\\right)[\/latex]<\/p>\n

                                                                                                                          [latex]-75+25[\/latex]<\/p>\n

                                                                                                                          \u221250<\/p>\n

                                                                                                                          [latex]54+\\left(-28\\right)[\/latex]<\/p>\n

                                                                                                                          [latex]11+\\left(-15\\right)+3[\/latex]<\/p>\n

                                                                                                                          \u22121<\/p>\n

                                                                                                                          [latex]-19+\\left(-42\\right)+12[\/latex]<\/p>\n

                                                                                                                          [latex]-3+6\\left(-1+5\\right)[\/latex]<\/p>\n

                                                                                                                          21<\/p>\n

                                                                                                                          [latex]10+4\\left(-3+7\\right)[\/latex]<\/p>\n

                                                                                                                          Evaluate Variable Expressions with Integers<\/strong><\/h4>\n

                                                                                                                          In the following exercises, evaluate each expression.<\/p>\n

                                                                                                                          [latex]n+4\\text{when}[\/latex]<\/p>\n

                                                                                                                          \u24d0 [latex]n=-1[\/latex]
                                                                                                                          \n\u24d1 [latex]n=-20[\/latex]<\/p>\n

                                                                                                                            \n
                                                                                                                          1. \u24d0 3<\/li>\n
                                                                                                                          2. \u24d1 \u221216<\/li>\n<\/ol>\n

                                                                                                                            [latex]x+\\left(-9\\right)\\text{when}[\/latex]<\/p>\n

                                                                                                                            \u24d0 [latex]x=3[\/latex]
                                                                                                                            \n\u24d1 [latex]x=-3[\/latex]<\/p>\n

                                                                                                                            [latex]{\\left(x+y\\right)}^{3}\\text{when}x=-4,y=1[\/latex]<\/p>\n

                                                                                                                            \u221227<\/p>\n

                                                                                                                            [latex]{\\left(u+v\\right)}^{2}\\text{when}u=-4,v=11[\/latex]<\/p>\n

                                                                                                                             <\/p>\n<\/div>\n

                                                                                                                             <\/p>\n

                                                                                                                            \n

                                                                                                                            TRANSLATE WORD PHRASES TO ALGEBRAIC EXPRESSIONS<\/h3>\n

                                                                                                                            In the following exercises, translate each phrase into an algebraic expression and then simplify.<\/p>\n

                                                                                                                            [latex]\\text{the sum of -8 and 2}[\/latex]<\/p>\n

                                                                                                                            \u22128 + 2 = \u22126<\/p>\n

                                                                                                                            [latex]\\text{4 more than -12}[\/latex]<\/p>\n

                                                                                                                            [latex]\\text{10 more than the sum of -5 and -6}[\/latex]<\/p>\n

                                                                                                                            10 + [\u22125 + (\u22126)] = \u22121<\/p>\n

                                                                                                                            [latex]\\text{the sum of}3\\text{and}-5,\\text{increased by 18}[\/latex]<\/p>\n

                                                                                                                             <\/p>\n<\/div>\n

                                                                                                                             <\/p>\n

                                                                                                                            \n

                                                                                                                            Add Integers in Applications<\/strong><\/h3>\n

                                                                                                                            In the following exercises, solve.<\/p>\n

                                                                                                                            Temperature<\/strong><\/h4>\n

                                                                                                                            On Monday, the high temperature in Denver was [latex]\\text{-4 degrees.}[\/latex] Tuesday\u2019s high temperature was [latex]\\text{20 degrees}[\/latex] more. What was the high temperature on Tuesday?<\/p>\n

                                                                                                                            16 degrees<\/p>\n

                                                                                                                            Credit<\/strong><\/h4>\n

                                                                                                                            Frida owed [latex]{$75}[\/latex] on her credit card. Then she charged [latex]{$21}[\/latex] more. What was her new balance?<\/p>\n

                                                                                                                            Subtract Integers<\/p>\n

                                                                                                                            Model Subtraction of Integers<\/strong><\/h4>\n

                                                                                                                            In the following exercises, model the following.<\/p>\n

                                                                                                                            [latex]6 - 1[\/latex]<\/p>\n

                                                                                                                            \"This<\/p>\n

                                                                                                                            5<\/p>\n

                                                                                                                            [latex]-4-\\left(-3\\right)[\/latex]<\/p>\n

                                                                                                                            [latex]2-\\left(-5\\right)[\/latex]<\/p>\n

                                                                                                                            \"This<\/p>\n

                                                                                                                            7<\/p>\n

                                                                                                                            [latex]-1 - 4[\/latex]<\/p>\n

                                                                                                                            Simplify Expressions with Integers<\/strong><\/h4>\n

                                                                                                                            In the following exercises, simplify each expression.<\/p>\n

                                                                                                                            [latex]24 - 16[\/latex]<\/p>\n

                                                                                                                            8<\/p>\n

                                                                                                                            [latex]19-\\left(-9\\right)[\/latex]<\/p>\n

                                                                                                                            [latex]-31 - 7[\/latex]<\/p>\n

                                                                                                                            \u221238<\/p>\n

                                                                                                                            [latex]-40-\\left(-11\\right)[\/latex]<\/p>\n

                                                                                                                            [latex]-52-\\left(-17\\right)-23[\/latex]<\/p>\n

                                                                                                                            \u221258<\/p>\n

                                                                                                                            [latex]25-\\left(-3 - 9\\right)[\/latex]<\/p>\n

                                                                                                                            [latex]\\left(1 - 7\\right)-\\left(3 - 8\\right)[\/latex]<\/p>\n

                                                                                                                            \u22121<\/p>\n

                                                                                                                            [latex]{3}^{2}-{7}^{2}[\/latex]<\/p>\n

                                                                                                                            Evaluate Variable Expressions with Integers<\/strong>
                                                                                                                            \nIn the following exercises, evaluate each expression.<\/p>\n

                                                                                                                            [latex]x - 7\\text{when}[\/latex]<\/p>\n

                                                                                                                            \u24d0 [latex]x=5[\/latex]
                                                                                                                            \n\u24d1 [latex]x=-4[\/latex]<\/p>\n

                                                                                                                              \n
                                                                                                                            1. \u24d0 \u22122<\/li>\n
                                                                                                                            2. \u24d1 \u221211<\/li>\n<\/ol>\n

                                                                                                                              [latex]10-y\\text{when}[\/latex]<\/p>\n

                                                                                                                              \u24d0 [latex]y=15[\/latex]
                                                                                                                              \n\u24d1 [latex]y=-16[\/latex]<\/p>\n

                                                                                                                              [latex]2{n}^{2}-n+5\\text{when}n=-4[\/latex]<\/p>\n

                                                                                                                              41<\/p>\n

                                                                                                                              [latex]-15 - 3{u}^{2}\\text{when}u=-5[\/latex]<\/p>\n

                                                                                                                              Translate Phrases to Algebraic Expressions<\/strong><\/h4>\n

                                                                                                                              In the following exercises, translate each phrase into an algebraic expression and then simplify.<\/p>\n

                                                                                                                              the difference of [latex]-12\\text{and}5[\/latex]<\/p>\n

                                                                                                                              \u221212 \u2212 5 = \u221217<\/p>\n

                                                                                                                              subtract [latex]23[\/latex] from [latex]-50[\/latex]<\/p>\n

                                                                                                                              Subtract Integers in Applications<\/strong><\/h4>\n

                                                                                                                              In the following exercises, solve the given applications.<\/p>\n

                                                                                                                              Temperature<\/strong><\/h4>\n

                                                                                                                              One morning the temperature in Bangor, Maine was [latex]\\text{18 degrees.}[\/latex] By afternoon, it had dropped [latex]\\text{20 degrees.}[\/latex] What was the afternoon temperature?<\/p>\n

                                                                                                                              \u22122 degrees<\/p>\n

                                                                                                                              Temperature<\/strong><\/h4>\n

                                                                                                                              On January 4, the high temperature in Laredo, Texas was [latex]\\text{78 degrees,}[\/latex] and the high in Houlton, Maine was [latex]-28\\text{degrees}[\/latex]. What was the difference in temperature of Laredo and Houlton?<\/p>\n

                                                                                                                               <\/p>\n<\/div>\n

                                                                                                                              \n

                                                                                                                              Multiply and Divide Integers<\/h3>\n

                                                                                                                              Multiply Integers<\/strong>
                                                                                                                              \nIn the following exercises, multiply.<\/p>\n

                                                                                                                              [latex]-9\\cdot 4[\/latex]<\/p>\n

                                                                                                                              \u221236<\/p>\n

                                                                                                                              [latex]5\\left(-7\\right)[\/latex]<\/p>\n

                                                                                                                              [latex]\\left(-11\\right)\\left(-11\\right)[\/latex]<\/p>\n

                                                                                                                              121<\/p>\n

                                                                                                                              [latex]-1\\cdot 6[\/latex]<\/p>\n

                                                                                                                              Divide Integers<\/strong>
                                                                                                                              \nIn the following exercises, divide.<\/p>\n

                                                                                                                              [latex]56\\div \\left(-8\\right)[\/latex]<\/p>\n

                                                                                                                              \u22127<\/p>\n

                                                                                                                              [latex]-120\\div \\left(-6\\right)[\/latex]<\/p>\n

                                                                                                                              [latex]-96\\div 12[\/latex]<\/p>\n

                                                                                                                              \u22128<\/p>\n

                                                                                                                              [latex]96\\div \\left(-16\\right)[\/latex]<\/p>\n

                                                                                                                              [latex]45\\div \\left(-1\\right)[\/latex]<\/p>\n

                                                                                                                              \u221245<\/p>\n

                                                                                                                              [latex]-162\\div \\left(-1\\right)[\/latex]<\/p>\n

                                                                                                                               <\/p>\n<\/div>\n

                                                                                                                               <\/p>\n

                                                                                                                              \n

                                                                                                                              Simplify Expressions with Integers<\/strong><\/h3>\n

                                                                                                                              In the following exercises, simplify each expression.<\/p>\n

                                                                                                                              [latex]5\\left(-9\\right)-3\\left(-12\\right)[\/latex]<\/p>\n

                                                                                                                              \u22129<\/p>\n

                                                                                                                              [latex]{\\left(-2\\right)}^{5}[\/latex]<\/p>\n

                                                                                                                              [latex]-{3}^{4}[\/latex]<\/p>\n

                                                                                                                              \u221281<\/p>\n

                                                                                                                              [latex]\\left(-3\\right)\\left(4\\right)\\left(-5\\right)\\left(-6\\right)[\/latex]<\/p>\n

                                                                                                                              [latex]42 - 4\\left(6 - 9\\right)[\/latex]<\/p>\n

                                                                                                                              54<\/p>\n

                                                                                                                              [latex]\\left(8 - 15\\right)\\left(9 - 3\\right)[\/latex]<\/p>\n

                                                                                                                              [latex]-2\\left(-18\\right)\\div 9[\/latex]<\/p>\n

                                                                                                                              4<\/p>\n

                                                                                                                              [latex]45\\div \\left(-3\\right)-12[\/latex]<\/p>\n

                                                                                                                               <\/p>\n<\/div>\n

                                                                                                                               <\/p>\n

                                                                                                                              \n

                                                                                                                              Evaluate Variable Expressions with Integers<\/strong><\/h3>\n

                                                                                                                              In the following exercises, evaluate each expression.<\/p>\n

                                                                                                                              [latex]7x - 3\\text{when}x=-9[\/latex]<\/p>\n

                                                                                                                              \u221266<\/p>\n

                                                                                                                              [latex]16 - 2n\\text{when}n=-8[\/latex]<\/p>\n

                                                                                                                              [latex]5a+8b\\text{when}a=-2,b=-6[\/latex]<\/p>\n

                                                                                                                              \u221258<\/p>\n

                                                                                                                              [latex]{x}^{2}+5x+4\\text{when}x=-3[\/latex]<\/p>\n

                                                                                                                               <\/p>\n

                                                                                                                               <\/p>\n<\/div>\n

                                                                                                                               <\/p>\n

                                                                                                                              \n

                                                                                                                              Translate Word Phrases to Algebraic Expressions<\/strong><\/h3>\n

                                                                                                                              In the following exercises, translate to an algebraic expression and simplify if possible.<\/p>\n

                                                                                                                              the product of [latex]-12[\/latex] and [latex]6[\/latex]<\/p>\n

                                                                                                                              \u221212(6) = \u221272<\/p>\n

                                                                                                                              the quotient of [latex]3[\/latex] and the sum of [latex]-7[\/latex] and [latex]s[\/latex]<\/p>\n

                                                                                                                              Solve Equations using Integers; The Division Property of Equality
                                                                                                                              \nDetermine Whether a Number is a Solution of an Equation<\/strong>
                                                                                                                              \nIn the following exercises, determine whether each number is a solution of the given equation.<\/p>\n

                                                                                                                              [latex]5x - 10=-35[\/latex]<\/p>\n

                                                                                                                              \u24d0 [latex]x=-9[\/latex]
                                                                                                                              \n\u24d1 [latex]x=-5[\/latex]
                                                                                                                              \n\u24d2 [latex]x=5[\/latex]<\/p>\n

                                                                                                                                \n
                                                                                                                              1. \u24d0 no<\/li>\n
                                                                                                                              2. \u24d1 yes<\/li>\n
                                                                                                                              3. \u24d2 no<\/li>\n<\/ol>\n

                                                                                                                                [latex]8u+24=-32[\/latex]<\/p>\n

                                                                                                                                \u24d0 [latex]u=-7[\/latex]
                                                                                                                                \n\u24d1 [latex]u=-1[\/latex]
                                                                                                                                \n\u24d2 [latex]u=7[\/latex]<\/p>\n

                                                                                                                                Using the Addition and Subtraction Properties of Equality<\/strong>
                                                                                                                                \nIn the following exercises, solve.<\/p>\n

                                                                                                                                [latex]a+14=2[\/latex]<\/p>\n

                                                                                                                                \u221212<\/p>\n

                                                                                                                                [latex]b - 9=-15[\/latex]<\/p>\n

                                                                                                                                \u22126<\/p>\n

                                                                                                                                [latex]c+\\left(-10\\right)=-17[\/latex]<\/p>\n

                                                                                                                                \u22127<\/p>\n

                                                                                                                                [latex]d-\\left(-6\\right)=-26[\/latex]<\/p>\n

                                                                                                                                \u221232<\/p>\n

                                                                                                                                 <\/p>\n<\/div>\n

                                                                                                                                 <\/p>\n

                                                                                                                                \n

                                                                                                                                Model the Division Property of Equality<\/strong><\/h3>\n

                                                                                                                                In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.<\/p>\n

                                                                                                                                \"This<\/p>\n

                                                                                                                                3x<\/em> = 9; x<\/em> = 3><\/p>\n

                                                                                                                                \"This<\/p>\n

                                                                                                                                Solve Equations Using the Division Property of Equality<\/strong>
                                                                                                                                \nIn the following exercises, solve each equation using the division property of equality and check the solution.<\/p>\n

                                                                                                                                [latex]8p=72[\/latex]<\/p>\n

                                                                                                                                9<\/p>\n

                                                                                                                                [latex]-12q=48[\/latex]<\/p>\n

                                                                                                                                [latex]-16r=-64[\/latex]<\/p>\n

                                                                                                                                4<\/p>\n

                                                                                                                                [latex]-5s=-100[\/latex]<\/p>\n

                                                                                                                                Translate to an Equation and Solve.<\/strong>
                                                                                                                                \nIn the following exercises, translate and solve.<\/p>\n

                                                                                                                                [latex]\\text{The product of -6 and}y\\text{is}-42[\/latex]<\/p>\n

                                                                                                                                \u22126y<\/em> = \u221242; y<\/em> = 7<\/p>\n

                                                                                                                                [latex]\\text{The difference of}z\\text{and -13 is -18.}[\/latex]<\/p>\n

                                                                                                                                Four more than [latex]m[\/latex] is [latex]-48[\/latex].<\/p>\n

                                                                                                                                m<\/em> + 4 = \u221248; m<\/em> = \u221252<\/p>\n

                                                                                                                                [latex]\\text{The product of -21 and}n\\text{is 63.}[\/latex]<\/p>\n

                                                                                                                                <\/h2>\n<\/div>\n

                                                                                                                                 <\/p>\n

                                                                                                                                \n

                                                                                                                                Everyday Math<\/h3>\n

                                                                                                                                Describe how you have used two topics from this chapter in your life outside of your math class during the past month.<\/p>\n<\/div>\n

                                                                                                                                 <\/p>\n

                                                                                                                                \n

                                                                                                                                Chapter Practice Test<\/h3>\n

                                                                                                                                Locate and label [latex]0,2,-4[\/latex], and [latex]-1[\/latex] on a number line.<\/p>\n

                                                                                                                                In the following exercises, compare the numbers, using [latex]<\\text{or}>\\text{or}\\text{=.}[\/latex]<\/p>\n

                                                                                                                                  \n
                                                                                                                                1. \u24d0 [latex]-6\\text{__}3[\/latex]<\/li>\n
                                                                                                                                2. \u24d1 [latex]-1\\text{__}-4[\/latex]<\/li>\n<\/ol>\n
                                                                                                                                    \n
                                                                                                                                  1. \u24d0 <<\/li>\n
                                                                                                                                  2. \u24d1 ><\/li>\n<\/ol>\n
                                                                                                                                      \n
                                                                                                                                    1. \u24d0 [latex]-5\\text{__}|-5|[\/latex]<\/li>\n
                                                                                                                                    2. \u24d1 [latex]-|-2|\\text{__}-2[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                      In the following exercises, find the opposite of each number.<\/p>\n

                                                                                                                                        \n
                                                                                                                                      1. \u24d0 [latex]-7[\/latex]<\/li>\n
                                                                                                                                      2. \u24d1 [latex]8[\/latex]<\/li>\n<\/ol>\n
                                                                                                                                          \n
                                                                                                                                        1. \u24d0 7<\/li>\n
                                                                                                                                        2. \u24d1 \u22128<\/li>\n<\/ol>\n

                                                                                                                                          In the following exercises, simplify.<\/p>\n

                                                                                                                                          [latex]-\\left(-22\\right)[\/latex]<\/p>\n

                                                                                                                                          [latex]|4 - 9|[\/latex]<\/p>\n

                                                                                                                                          5<\/p>\n

                                                                                                                                          [latex]-8+6[\/latex]<\/p>\n

                                                                                                                                          [latex]-15+\\left(-12\\right)[\/latex]<\/p>\n

                                                                                                                                          \u221227<\/p>\n

                                                                                                                                          [latex]-7-\\left(-3\\right)[\/latex]<\/p>\n

                                                                                                                                          [latex]10-\\left(5 - 6\\right)[\/latex]<\/p>\n

                                                                                                                                          11<\/p>\n

                                                                                                                                          [latex]-3\\cdot 8[\/latex]<\/p>\n

                                                                                                                                          [latex]-6\\left(-9\\right)[\/latex]<\/p>\n

                                                                                                                                          54<\/p>\n

                                                                                                                                          [latex]70\\div \\left(-7\\right)[\/latex]<\/p>\n

                                                                                                                                          [latex]{\\left(-2\\right)}^{3}[\/latex]<\/p>\n

                                                                                                                                          \u22128<\/p>\n

                                                                                                                                          [latex]-{4}^{2}[\/latex]<\/p>\n

                                                                                                                                          [latex]16 - 3\\left(5 - 7\\right)[\/latex]<\/p>\n

                                                                                                                                          22<\/p>\n

                                                                                                                                          [latex]|21 - 6|-|-8|[\/latex]<\/p>\n

                                                                                                                                          In the following exercises, evaluate.<\/p>\n

                                                                                                                                          [latex]35-a\\text{when}a=-4[\/latex]<\/p>\n

                                                                                                                                          39<\/p>\n

                                                                                                                                          [latex]{\\left(-2r\\right)}^{2}\\text{when}r=3[\/latex]<\/p>\n

                                                                                                                                          [latex]3m - 2n\\text{when}m=6,n=-8[\/latex]<\/p>\n

                                                                                                                                          34<\/p>\n

                                                                                                                                          [latex]-|-y|\\text{when}y=17[\/latex]<\/p>\n

                                                                                                                                          In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.<\/p>\n

                                                                                                                                          the difference of \u22127 and \u22124<\/p>\n

                                                                                                                                          \u22127 \u2212 (\u22124) = \u22123<\/p>\n

                                                                                                                                          the quotient of [latex]25[\/latex] and the sum of [latex]m[\/latex] and [latex]n[\/latex].<\/p>\n

                                                                                                                                          In the following exercises, solve.<\/p>\n

                                                                                                                                          Early one morning, the temperature in Syracuse was [latex]-8\\text{^\\circ F.}[\/latex] By noon, it had risen [latex]\\text{12^\\circ .}[\/latex] What was the temperature at noon?<\/p>\n

                                                                                                                                          4\u00b0F<\/p>\n

                                                                                                                                          Collette owed [latex]{$128}[\/latex] on her credit card. Then she charged [latex]{$65.}[\/latex] What was her new balance?<\/p>\n

                                                                                                                                          In the following exercises, solve.<\/p>\n

                                                                                                                                          [latex]n+6=5[\/latex]<\/p>\n

                                                                                                                                          n<\/em> = \u22121<\/p>\n

                                                                                                                                          [latex]p - 11=-4[\/latex]<\/p>\n

                                                                                                                                          [latex]-9r=-54[\/latex]<\/p>\n

                                                                                                                                          r<\/em> = 6<\/p>\n

                                                                                                                                          In the following exercises, translate and solve.<\/p>\n

                                                                                                                                          [latex]\\text{The product of 15 and}x\\text{is 75.}[\/latex]<\/p>\n

                                                                                                                                          [latex]\\text{Eight less than}y\\text{is -32.}[\/latex]<\/p>\n

                                                                                                                                          y<\/em> \u2212 8 = \u221232; y<\/em> = \u221224<\/p>\n<\/div>\n\n\t\t\t

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