{"id":6639,"date":"2018-03-06T19:45:18","date_gmt":"2018-03-06T19:45:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-albany-chemistry\/?post_type=chapter&p=6639"},"modified":"2018-06-04T15:26:14","modified_gmt":"2018-06-04T15:26:14","slug":"inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-albany-chemistry\/chapter\/inequalities\/","title":{"raw":"0.6 Inequalities","rendered":"0.6 Inequalities"},"content":{"raw":"
\r\n\r\n[embed]https:\/\/youtu.be\/ecjuK5swjqQ[\/embed]\r\n

QUICK REFERENCE<\/h3>\r\nInequality Signs<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Symbol<\/th>\r\nWords<\/th>\r\nExample<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]\\neq [\/latex]<\/td>\r\nnot equal to<\/td>\r\n[latex]{2}\\neq{8}[\/latex], 2<\/i>\u00a0is<\/strong> not equal<\/b> to 8.<\/em><\/td>\r\n<\/tr>\r\n
[latex]\\gt[\/latex]<\/td>\r\ngreater than<\/td>\r\n[latex]{5}\\gt{1}[\/latex], 5<\/i>\u00a0is greater than<\/strong>\u00a01<\/i><\/td>\r\n<\/tr>\r\n
[latex]\\lt[\/latex]<\/td>\r\nless than<\/td>\r\n[latex]{2}\\lt{11}[\/latex], 2\u00a0<\/i>is less than<\/b>\u00a011<\/i><\/td>\r\n<\/tr>\r\n
[latex] \\geq [\/latex]<\/td>\r\ngreater than or equal to<\/td>\r\n[latex]{4}\\geq{ 4}[\/latex], 4\u00a0<\/i>is greater than or equal to<\/b>\u00a04<\/i><\/td>\r\n<\/tr>\r\n
[latex]\\leq [\/latex]<\/td>\r\nless than or equal to<\/td>\r\n[latex]{7}\\leq{9}[\/latex], 7<\/i>\u00a0is less than or equal to<\/b>\u00a09<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAddition and Subtraction Properties of Inequalities<\/strong>\r\n

If [latex]a>b[\/latex],\u00a0<\/i>then [latex]a+c>b+c[\/latex].<\/p>\r\n

If\u00a0[latex]a>b[\/latex], <\/i>then [latex]a\u2212c>b\u2212c[\/latex].<\/p>\r\nMultiplication and Division Properties of Inequality<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n
Start With<\/strong><\/td>\r\nMultiply By<\/strong><\/td>\r\nFinal Inequality<\/strong><\/td>\r\n<\/tr>\r\n
[latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex]<\/td>\r\n[latex]ac>bc[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]a>b[\/latex]<\/td>\r\n[latex]-c[\/latex]<\/td>\r\n[latex]ac<bc[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
Start With<\/strong><\/td>\r\nDivide By<\/strong><\/td>\r\nFinal Inequality<\/strong><\/td>\r\n<\/tr>\r\n
[latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex]<\/td>\r\n[latex] \\displaystyle \\frac{a}{c}>\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]a>b[\/latex]<\/td>\r\n[latex]-c[\/latex]<\/td>\r\n[latex] \\displaystyle \\frac{a}{c}<\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n \r\n\r\n<\/div>\r\n

Inequalities<\/h2>\r\nFirst, let's define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right\u2014just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.\r\n
    \r\n \t
  • [latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than 9. Would you rather write\u00a0[latex]{x}\\lt{9}[\/latex] or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)<\/li>\r\n \t
  • [latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\r\n<\/ul>\r\nNote how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.\r\n\r\nFor example:\r\n
      \r\n \t
    • [latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\r\n \t
    • [latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex] note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.<\/li>\r\n<\/ul>\r\n

      Inequality Signs<\/h3>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it's easy to get tangled up in inequalities, just remember to read them from left to right.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Symbol<\/th>\r\nWords<\/th>\r\nExample<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      [latex]\\neq [\/latex]<\/td>\r\nnot equal to<\/td>\r\n[latex]{2}\\neq{8}[\/latex], 2<\/i>\u00a0is<\/strong> not equal<\/b> to 8.<\/em><\/td>\r\n<\/tr>\r\n
      [latex]\\gt[\/latex]<\/td>\r\ngreater than<\/td>\r\n[latex]{5}\\gt{1}[\/latex], 5<\/i>\u00a0is greater than<\/strong>\u00a01<\/i><\/td>\r\n<\/tr>\r\n
      [latex]\\lt[\/latex]<\/td>\r\nless than<\/td>\r\n[latex]{2}\\lt{11}[\/latex], 2\u00a0<\/i>is less than<\/b>\u00a011<\/i><\/td>\r\n<\/tr>\r\n
      [latex] \\geq [\/latex]<\/td>\r\ngreater than or equal to<\/td>\r\n[latex]{4}\\geq{ 4}[\/latex], 4\u00a0<\/i>is greater than or equal to<\/b>\u00a04<\/i><\/td>\r\n<\/tr>\r\n
      [latex]\\leq [\/latex]<\/td>\r\nless than or equal to<\/td>\r\n[latex]{7}\\leq{9}[\/latex], 7<\/i>\u00a0is less than or equal to<\/b>\u00a09<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe inequality [latex]x>y[\/latex]\u00a0can also be written as [latex]{y}<{x}[\/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.\r\n

      Solve Single-Step Inequalities<\/h2>\r\n

      Solve inequalities with addition and subtraction<\/h3>\r\nYou can solve most inequalities using inverse operations as you did for solving equations. \u00a0This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.\r\n
      \r\n

      Addition and Subtraction Properties of Inequality<\/h3>\r\nIf [latex]a>b[\/latex],\u00a0<\/i>then [latex]a+c>b+c[\/latex].\r\n\r\nIf\u00a0[latex]a>b[\/latex], <\/i>then [latex]a\u2212c>b\u2212c[\/latex].\r\n\r\n<\/div>\r\n
      \r\n

      Example<\/h3>\r\nSolve for x.<\/i>\r\n

      [latex] {x}+3\\lt{5}[\/latex]<\/p>\r\n[reveal-answer q=\"952771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"952771\"]\r\n\r\nIt is helpful to think of this inequality as asking you to find all the values for x<\/em>, including negative numbers, such that when you add three you will get a number less than 5.\r\n

      [latex] \\displaystyle \\begin{array}{l}x+3<\\,\\,\\,\\,5\\\\\\underline{\\,\\,\\,\\,\\,-3\\,\\,\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\,\\,2\\,\\,\\end{array}[\/latex]<\/p>\r\nIsolate the variable by subtracting 3 from both sides of the inequality.\r\n

      Answer<\/h4>\r\nInequality: \u00a0[latex]x<2[\/latex]\r\n\r\nInterval: \u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]\r\n\r\nGraph: \"Number\r\n\r\n<\/div>\r\nJust as you can check the solution to an equation, you can check a solution to an inequality. First, you check the endpoint by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions.\r\n\r\nThe example below shows how you could check that [latex]x<2[\/latex]\u00a0<\/i>is the solution to [latex]x+3<5[\/latex].<\/i>\r\n
      \r\n

      Example<\/h3>\r\nCheck that [latex]x<2[\/latex]\u00a0<\/i>is the solution to [latex]x+3<5[\/latex].\r\n\r\n[reveal-answer q=\"811564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811564\"]\r\n\r\nSubstitute the end point 2 into the related equation, [latex]x+3=5[\/latex].\r\n

      [latex]\\begin{array}{r}x+3=5 \\\\ 2+3=5 \\\\ 5=5\\end{array}[\/latex]<\/p>\r\nPick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)\r\n

      \"Number<\/p>\r\n

      [latex] \\displaystyle \\begin{array}{r}x+3<5 \\\\ 0+3<5 \\\\ 3<5\\end{array}[\/latex]<\/p>\r\nIt checks!\r\n\r\n[latex]x<2[\/latex] is the solution to [latex]x+3<5[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following examples show inequality problems that include operations with negative numbers.\r\n

      \r\n

      Example<\/h3>\r\nSolve for x<\/em>:\u00a0[latex]x-10\\leq-12[\/latex]\r\n[reveal-answer q=\"815894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815894\"]\r\n\r\nIsolate the variable by adding 10 to both sides of the inequality.\r\n

      [latex] \\displaystyle \\begin{array}{r}x-10\\le -12\\\\\\underline{\\,\\,\\,+10\\,\\,\\,\\,\\,+10}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,-2\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality: [latex]x\\leq-2[\/latex]\r\nInterval: [latex]\\left(-\\infty,-2\\right][\/latex]\r\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d [latex]\\left(\\leq\\right)[\/latex]. The blue arrow is drawn to the left of the point [latex]\u22122[\/latex] because these are the values that are less than [latex]\u22122[\/latex].\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the\u00a0solution to [latex]x-10\\leq -12[\/latex]\r\n[reveal-answer q=\"268062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"268062\"]\r\n\r\nSubstitute the end point [latex]\u22122[\/latex] into the related equation \u00a0[latex]x-10=\u221212[\/latex]\r\n

      [latex] \\displaystyle \\begin{array}{r}x-10=-12\\,\\,\\,\\\\\\text{Does}\\,\\,\\,-2-10=-12?\\\\-12=-12\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value less than [latex]\u22122[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)\r\n

      [latex] \\displaystyle \\begin{array}{r}x-10\\le -12\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,-5-10\\le -12?\\\\-15\\le -12\\,\\,\\,\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\r\n[latex]x\\leq -2[\/latex]\u00a0is the solution to [latex]x-10\\leq -12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      \r\n

      Example<\/h3>\r\nSolve for a<\/em>. [latex]a-17>-17[\/latex]\r\n[reveal-answer q=\"343031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343031\"]\r\n\r\nIsolate the variable by adding 17 to both sides of the inequality.\r\n

      [latex] \\displaystyle \\begin{array}{r}a-17>-17\\\\\\underline{\\,\\,\\,+17\\,\\,\\,\\,\\,+17}\\\\a\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\,\\,\\,\\,\\,0\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality:\u00a0[latex] \\displaystyle a\\,\\,>\\,0[\/latex]\r\n\r\nInterval: [latex]\\left(0,\\infty\\right)[\/latex] \u00a0Note how we use parentheses on the left to show that the solution does not include 0.\r\n\r\nGraph: Note the open circle to show that the solution does not include 0.\r\n
      \r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution to\u00a0[latex]a-17>-17[\/latex]\r\n[reveal-answer q=\"653357\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"653357\"]\r\n\r\nIs\u00a0[latex] \\displaystyle a\\,\\,>\\,0[\/latex] the correct solution to\u00a0\u00a0[latex]a-17>-17[\/latex]?\r\n\r\nSubstitute the end point 0 into the related equation.\r\n

      [latex] \\displaystyle \\begin{array}{r}a-17=-17\\,\\,\\,\\\\\\text{Does}\\,\\,\\,0-17=-17?\\\\-17=-17\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value greater than 0, such as 20, to check in the inequality. (This value will be on the shaded part of the graph.)\r\n

      [latex] \\displaystyle \\begin{array}{r}a-17>-17\\,\\,\\,\\\\\\text{Is }\\,\\,20-17>-17?\\\\3>-17\\,\\,\\,\\\\\\\\\\text{It checks!}\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex] \\displaystyle a\\,>\\,0[\/latex] is the solution to\u00a0[latex]a-17>-17[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWhat would you do if the variable were on the right side of the inequality? \u00a0In the following example, you will see how to handle this scenario.\r\n

      \r\n

      Example<\/h3>\r\nSolve for x<\/em>:\u00a0[latex]4\\geq{x}+5[\/latex]\r\n[reveal-answer q=\"815893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815893\"]\r\n\r\nIsolate the variable by adding 10 to both sides of the inequality.\r\n

      [latex] \\displaystyle \\begin{array}{r}4\\geq{x}+5 \\\\\\underline{\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\,\\,\\,x\\end{array}[\/latex]<\/p>\r\n

      Rewrite the inequality with the variable on the left - this makes writing the interval and drawing the graph easier.<\/p>\r\n

      [latex]x\\le{-1}[\/latex]<\/p>\r\n

      Note how the the pointy part of the inequality is still directed at the variable, so instead of reading as negative one is greater or equal to x, it now reads as x is less than or equal to negative one.<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality: [latex]x\\le{-1}[\/latex] This can also be written as\r\nInterval: [latex]\\left(-\\infty,-1\\right][\/latex]\r\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d . The blue arrow is drawn to the left of the point [latex]\u22121[\/latex] because these are the values that are less than [latex]\u22121[\/latex].\r\n\"(-oo,-1]\"\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the\u00a0solution to [latex]4\\geq{x}+5[\/latex]\r\n[reveal-answer q=\"568062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568062\"]\r\n\r\nSubstitute the end point [latex]\u22121[\/latex] into the related equation \u00a0[latex]4=x+5[\/latex]\r\n

      [latex] \\displaystyle \\begin{array}{r}4=x+5\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4=-1+5?\\\\-1=-1\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value less than [latex]\u22121[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)\r\n

      [latex] \\displaystyle \\begin{array}{r}4\\geq{-5}+5\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,4\\ge 0?\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\r\n[latex]x\\le{-1}[\/latex] is the solution to [latex]4\\geq{x}+5[\/latex][\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n

      Solve inequalities with multiplication and division<\/h3>\r\nSolving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let\u2019s look at what happens to the inequality when you multiply or divide each side by the same number.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Let's start with the true statement:\r\n[latex]10>5[\/latex]<\/td>\r\nLet's try again by starting with the same true statement:\r\n[latex]10>5[\/latex]<\/td>\r\n<\/tr>\r\n
      Next, multiply both sides by the same positive number:\r\n[latex]10\\cdot 2>5\\cdot 2[\/latex]<\/td>\r\nThis time, multiply both sides by the same negative number:\r\n[latex]10\\cdot-2>5 \\\\ \\,\\,\\,\\,\\,\\cdot -2\\,\\cdot-2[\/latex]<\/td>\r\n<\/tr>\r\n
      20 is greater than 10, so you still have a true inequality:\r\n[latex]20>10[\/latex]<\/td>\r\nWait a minute! [latex]\u221220[\/latex] is not <\/i>greater than [latex]\u221210[\/latex], so you have an untrue statement.\r\n[latex]\u221220>\u221210[\/latex]<\/td>\r\n<\/tr>\r\n
      When you multiply by a positive number, leave the inequality sign as it is!<\/td>\r\nYou must \u201creverse\u201d the inequality sign to make the statement true:\r\n[latex]\u221220<\u221210[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n\r\n\"Caution\"Caution! \u00a0When you multiply or divide by a negative number, \u201creverse\u201d the inequality sign. \u00a0 Whenever you multiply or divide both sides of an inequality by a negative\u00a0number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.\r\n\r\n<\/div>\r\n\r\n
      \r\n\r\n
      \r\n

      Multiplication and Division Properties of Inequality<\/h3>\r\n\r\n\r\n\r\n\r\n\r\n
      Start With<\/strong><\/td>\r\nMultiply By<\/strong><\/td>\r\nFinal Inequality<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex]<\/td>\r\n[latex]ac>bc[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]a>b[\/latex]<\/td>\r\n[latex]-c[\/latex]<\/td>\r\n[latex]ac<bc[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
      Start With<\/strong><\/td>\r\nDivide By<\/strong><\/td>\r\nFinal Inequality<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex]<\/td>\r\n[latex] \\displaystyle \\frac{a}{c}>\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]a>b[\/latex]<\/td>\r\n[latex]-c[\/latex]<\/td>\r\n[latex] \\displaystyle \\frac{a}{c}<\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n<\/div>\r\nKeep in mind that you only change the sign when you are multiplying and dividing by a negative<\/i> number. If you add or subtract<\/em> by a negative\u00a0number, the inequality stays the same.\r\n
      \r\n

      Example<\/h3>\r\nSolve for x.\u00a0<\/i>[latex]3x>12[\/latex]\r\n\r\n[reveal-answer q=\"691711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"691711\"]Divide both sides by 3 to isolate the variable.\r\n

      [latex] \\displaystyle \\begin{array}{r}\\underline{3x}>\\underline{12}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\\\x>4\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck your solution by first checking the end point 4, and then checking another solution for the inequality.\r\n

      [latex]\\begin{array}{r}3\\cdot4=12\\\\12=12\\\\3\\cdot10>12\\\\30>12\\\\\\text{It checks!}\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n

      Inequality: [latex] \\displaystyle x>4[\/latex]<\/p>\r\n

      Interval: [latex]\\left(4,\\infty\\right)[\/latex]<\/p>\r\n

      Graph: \"Number<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThere was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive<\/i> 3. In the next example, there is division by a negative number, so there is an additional step in the solution.\r\n

      \r\n

      Example<\/h3>\r\nSolve for x<\/em>. [latex]\u22122x>6[\/latex]\r\n\r\n[reveal-answer q=\"604033\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604033\"]Divide each side of the inequality by [latex]\u22122[\/latex] to isolate the variable, and change the direction of the inequality sign because of the division by a negative number.\r\n

      [latex] \\displaystyle \\begin{array}{r}\\underline{-2x}<\\underline{\\,6\\,}\\\\-2\\,\\,\\,\\,-2\\,\\\\x<-3\\end{array}[\/latex]<\/p>\r\nCheck your solution by first checking the end point [latex]\u22123[\/latex], and then checking another solution for the inequality.\r\n

      [latex]\\begin{array}{r}-2\\left(-3\\right)=6 \\\\6=6\\\\ -2\\left(-6\\right)>6 \\\\ 12>6\\end{array}[\/latex]<\/p>\r\nIt checks!\r\n

      Answer<\/h4>\r\nInequality: [latex] \\displaystyle x<-3[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty, -3\\right)[\/latex]\r\n\r\nGraph: \"Number\r\nBecause both sides of the inequality were divided by a negative number, [latex]\u22122[\/latex], the inequality symbol was switched from > to <.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      Combine properties of inequality to solve algebraic\u00a0inequalities<\/h2>\r\nA popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and\/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other.\r\n
      \r\n

      Example<\/h3>\r\nSolve for p<\/i>. [latex]4p+5<29[\/latex]\r\n\r\n[reveal-answer q=\"211828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211828\"]\r\n\r\nBegin to isolate the variable by subtracting 5 from both sides of the inequality.\r\n

      [latex] \\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\4p\\,\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,24\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by 4 to express the variable with a coefficient of 1.\r\n

      [latex]\\begin{array}{l}\\underline{4p}\\,<\\,\\,\\underline{24}\\,\\,\\\\\\,4\\,\\,\\,\\,<\\,\\,4\\\\\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality:\u00a0[latex]p<6[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,6\\right)[\/latex]\r\n\r\nGraph: Note the\u00a0open circle at the end point 6 to show that solutions to the inequality do not include 6.\u00a0The values where p<\/i> is less than 6 are found all along the number line to the left of 6.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"291597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"291597\"]\r\n\r\nCheck the end point 6 in the related equation.\r\n

      [latex] \\displaystyle \\begin{array}{r}4p+5=29\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4(6)+5=29?\\\\24+5=29\\,\\,\\,\\\\29=29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nTry another value to check the inequality. Let\u2019s use [latex]p=0[\/latex].\r\n

      [latex] \\displaystyle \\begin{array}{r}4p+5<29\\,\\,\\,\\\\\\text{Is}\\,\\,\\,4(0)+5<29?\\\\0+5<29\\,\\,\\,\\\\5<29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]p<6[\/latex] is the solution to\u00a0[latex]4p+5<29[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      \r\n

      Example<\/h3>\r\nSolve for x<\/i>: \u00a0[latex]3x\u20137\\ge 41[\/latex]\r\n[reveal-answer q=\"238157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238157\"]\r\n\r\nBegin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.\r\n

      [latex] \\displaystyle \\begin{array}{l}3x-7\\ge 41\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+7\\,\\,\\,\\,+7}\\\\\\frac{3x}{3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{48}{3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 16\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality: [latex]x\\ge 16[\/latex]\r\n\r\nInterval: [latex]\\left[16,\\infty\\right)[\/latex]\r\n\r\nGraph:\u00a0To graph this inequality, you draw a closed circle at the end point 16 on the number line\u00a0to show that solutions include the value 16. The line continues to the right from 16 because all the numbers greater than 16 will also make the inequality\u00a0[latex]3x\u20137\\ge 41[\/latex] true.\r\n\"Closed\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"437341\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"437341\"]\r\n\r\nFirst, check the end point 16 in the related equation.\r\n

      [latex] \\displaystyle \\begin{array}{r}3x-7=41\\,\\,\\,\\\\\\text{Does}\\,\\,\\,3(16)-7=41?\\\\48-7=41\\,\\,\\,\\\\41=41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Let\u2019s use [latex]x = 20[\/latex].\r\n

      [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,3x-7\\ge 41\\,\\,\\,\\\\\\text{Is}\\,\\,\\,\\,\\,3(20)-7\\ge 41?\\\\60-7\\ge 41\\,\\,\\,\\\\53\\ge 41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.\r\n

      \r\n

      Example<\/h3>\r\nSolve for p<\/i>. [latex]\u221258>14\u22126p[\/latex]\r\n\r\n[reveal-answer q=\"424351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"424351\"]\r\n\r\nNote how the variable is on the right hand side of the inequality, the method for solving does not change in this case.\r\n\r\nBegin to isolate the variable by subtracting 14 from both sides of the inequality.\r\n

      [latex] \\displaystyle \\begin{array}{l}\u221258\\,\\,>14\u22126p\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-14\\,\\,\\,\\,\\,\\,\\,-14}\\\\-72\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>-6p\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by [latex]\u22126[\/latex] to express the variable with a coefficient of 1.\u00a0Dividing by a negative number results in reversing the inequality sign.\r\n

      [latex]\\begin{array}{l}\\underline{-72}>\\underline{-6p}\\\\-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\\\\\,\\,\\,\\,\\,\\,12\\lt{p}\\end{array}[\/latex]<\/p>\r\n

      We can also write this as [latex]p>12[\/latex]. \u00a0 Notice how the inequality sign is still opening up toward the variable p.<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality: [latex]p>12[\/latex]\r\nInterval: [latex]\\left(12,\\infty\\right)[\/latex]\r\nGraph: The graph of the inequality p <\/i>> 12 has an open circle at 12 with an arrow stretching to the right.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"500309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500309\"]\r\n\r\nFirst, check the end point 12 in the related equation.\r\n

      [latex]\\begin{array}{r}-58=14-6p\\\\-58=14-6\\left(12\\right)\\\\-58=14-72\\\\-58=-58\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Try 100.\r\n

      [latex]\\begin{array}{r}-58>14-6p\\\\-58>14-6\\left(100\\right)\\\\-58>14-600\\\\-58>-586\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      Simplify and solve algebraic inequalities using the distributive property<\/h2>\r\nAs with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.\r\n
      \r\n

      Example<\/h3>\r\nSolve for x<\/em>. [latex]2\\left(3x\u20135\\right)\\leq 4x+6[\/latex]\r\n\r\n[reveal-answer q=\"587737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587737\"]\r\n\r\nDistribute to clear the parentheses.\r\n

      [latex] \\displaystyle \\begin{array}{r}\\,2(3x-5)\\leq 4x+6\\\\\\,\\,\\,\\,6x-10\\leq 4x+6\\end{array}[\/latex]<\/p>\r\nSubtract 4x <\/i>from both sides to get the variable term on one side only.\r\n

      [latex]\\begin{array}{r}6x-10\\le 4x+6\\\\\\underline{-4x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-4x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,2x-10\\,\\,\\leq \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides to isolate the variable.\r\n

      [latex]\\begin{array}{r}\\\\\\,\\,\\,2x-10\\,\\,\\le \\,\\,\\,\\,\\,\\,\\,\\,6\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,+10\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\\\,\\,\\,2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,\\,\\,16\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides by 2 to express the variable with a coefficient of 1.\r\n

      [latex]\\begin{array}{r}\\underline{2x}\\le \\,\\,\\,\\underline{16}\\\\\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\le \\,\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\nInequality: [latex]x\\le8[\/latex]\r\nInterval: [latex]\\left(-\\infty,8\\right][\/latex]\r\nGraph: The graph of this solution set includes 8 and everything left of 8 on the number line.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"808701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808701\"]\r\n\r\nFirst, check the end point 8 in the related equation.\r\n

      [latex] \\displaystyle \\begin{array}{r}2(3x-5)=4x+6\\,\\,\\,\\,\\,\\,\\\\2(3\\,\\cdot \\,8-5)=4\\,\\cdot \\,8+6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(24-5)=32+6\\,\\,\\,\\,\\,\\,\\\\2(19)=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\38=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, choose another solution and evaluate the inequality for that value to make sure it is a true statement.\u00a0Try 0.\r\n

      [latex] \\displaystyle \\begin{array}{l}2(3\\,\\cdot \\,0-5)\\le 4\\,\\cdot \\,0+6?\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(-5)\\le 6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10\\le 6\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]x\\le8[\/latex] is the solution to\u00a0[latex]\\left(-\\infty,8\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      <\/h2>","rendered":"
      \n