{"id":437,"date":"2015-10-26T18:33:18","date_gmt":"2015-10-26T18:33:18","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=437"},"modified":"2015-11-12T18:37:59","modified_gmt":"2015-11-12T18:37:59","slug":"using-the-properties-of-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/using-the-properties-of-inequalities\/","title":{"raw":"Using the Properties of Inequalities","rendered":"Using the Properties of Inequalities"},"content":{"raw":"When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property<\/strong> and the multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.\r\n
\r\n

A General Note: Properties of Inequalities<\/h3>\r\n

[latex]\\begin{array}{ll}\\text{Addition Property}\\hfill& \\text{If }a< b,\\text{ then }a+c< b+c.\\hfill \\\\ \\hfill & \\hfill \\\\ \\text{Multiplication Property}\\hfill & \\text{If }a< b\\text{ and }c> 0,\\text{ then }ac< bc.\\hfill \\\\ \\hfill & \\text{If }a< b\\text{ and }c< 0,\\text{ then }ac> bc.\\hfill \\end{array}[\/latex]<\/p>\r\nThese properties also apply to [latex]a\\le b[\/latex], [latex]a>b[\/latex], and [latex]a\\ge b[\/latex].\r\n\r\n<\/div>\r\n

\r\n

Example 3: Demonstrating the Addition Property<\/h3>\r\nIllustrate the addition property for inequalities by solving each of the following:\r\n

a. [latex]x - 15<4[\/latex]\r\nb. [latex]6\\ge x - 1[\/latex]\r\nc. [latex]x+7>9[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

Solution<\/h3>\r\nThe addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.\r\n\r\na. [latex]\\begin{array}{ll}x - 15<4\\hfill & \\hfill \\\\ x - 15+15<4+15 \\hfill & \\text{Add 15 to both sides.}\\hfill \\\\ x<19\\hfill & \\hfill \\end{array}[\/latex]\r\n\r\nb. [latex]\\begin{array}{ll}6\\ge x - 1\\hfill & \\hfill \\\\ 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill & \\hfill \\end{array}[\/latex]\r\n\r\nc. [latex]\\begin{array}{ll}x+7>9\\hfill & \\hfill \\\\ x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill \\\\ x>2\\hfill & \\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n
\r\n

Try It 3<\/h3>\r\nSolve [latex]3x - 2<1[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
\r\n

Example 4: Demonstrating the Multiplication Property<\/h3>\r\nIllustrate the multiplication property for inequalities by solving each of the following:\r\n
    \r\n\t
  1. [latex]3x<6[\/latex]<\/li>\r\n\t
  2. [latex]-2x - 1\\ge 5[\/latex]<\/li>\r\n\t
  3. [latex]5-x>10[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\na. [latex]\\begin{array}{l}3x<6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)<\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x<2\\hfill \\end{array}[\/latex]\r\n\r\n \r\n
    b.\u00a0[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill & \\hfill \\\\ -2x\\ge 6\\hfill & \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill & \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\n \r\n
    c.\u00a0[latex]\\begin{array}{ll}5-x>10\\hfill & \\hfill \\\\ -x>5\\hfill & \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)\\hfill & \\text{Multiply by }-1.\\hfill \\\\ x<-5\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n
    \r\n

    Try It 4<\/h3>\r\nSolve [latex]4x+7\\ge 2x - 3[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

    Solving Inequalities in One Variable Algebraically<\/h2>\r\nAs the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.\r\n
    \r\n

    Example 5: Solving an Inequality Algebraically<\/h3>\r\nSolve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nSolving this inequality is similar to solving an equation up until the last step.\r\n
    [latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill & \\hfill \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill & \\text{Isolate the variable term}.\\hfill \\\\ x\\le 1\\hfill & \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.\r\n\r\n<\/div>\r\n
    \r\n

    Try It 5<\/h3>\r\nSolve the inequality and write the answer using interval notation: [latex]-x+4<\\frac{1}{2}x+1[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
    \r\n

    Example 6: Solving an Inequality with Fractions<\/h3>\r\nSolve the following inequality and write the answer in interval notation: [latex]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nWe begin solving in the same way we do when solving an equation.\r\n
    [latex]\\begin{array}{ll}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill & \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill & \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill & \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill & \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill & \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill & \\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is the interval [latex]\\left(-\\infty ,\\frac{15}{34}\\right][\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Try It 6<\/h3>\r\nSolve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

    When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property<\/strong> and the multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.<\/p>\n

    \n

    A General Note: Properties of Inequalities<\/h3>\n

    [latex]\\begin{array}{ll}\\text{Addition Property}\\hfill& \\text{If }a< b,\\text{ then }a+c< b+c.\\hfill \\\\ \\hfill & \\hfill \\\\ \\text{Multiplication Property}\\hfill & \\text{If }a< b\\text{ and }c> 0,\\text{ then }ac< bc.\\hfill \\\\ \\hfill & \\text{If }a< b\\text{ and }c< 0,\\text{ then }ac> bc.\\hfill \\end{array}[\/latex]<\/p>\n

    These properties also apply to [latex]a\\le b[\/latex], [latex]a>b[\/latex], and [latex]a\\ge b[\/latex].<\/p>\n<\/div>\n

    \n

    Example 3: Demonstrating the Addition Property<\/h3>\n

    Illustrate the addition property for inequalities by solving each of the following:<\/p>\n

    a. [latex]x - 15<4[\/latex]\nb. [latex]6\\ge x - 1[\/latex]\nc. [latex]x+7>9[\/latex]<\/p>\n<\/div>\n

    \n

    Solution<\/h3>\n

    The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.<\/p>\n

    a. [latex]\\begin{array}{ll}x - 15<4\\hfill & \\hfill \\\\ x - 15+15<4+15 \\hfill & \\text{Add 15 to both sides.}\\hfill \\\\ x<19\\hfill & \\hfill \\end{array}[\/latex]\n\nb. [latex]\\begin{array}{ll}6\\ge x - 1\\hfill & \\hfill \\\\ 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill & \\hfill \\end{array}[\/latex]\n\nc. [latex]\\begin{array}{ll}x+7>9\\hfill & \\hfill \\\\ x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill \\\\ x>2\\hfill & \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

    \n

    Try It 3<\/h3>\n

    Solve [latex]3x - 2<1[\/latex].\n\nSolution<\/a><\/p>\n<\/div>\n

    \n

    Example 4: Demonstrating the Multiplication Property<\/h3>\n

    Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\n

      \n
    1. [latex]3x<6[\/latex]<\/li>\n
    2. [latex]-2x - 1\\ge 5[\/latex]<\/li>\n
    3. [latex]5-x>10[\/latex]<\/li>\n<\/ol>\n<\/div>\n
      \n

      Solution<\/h3>\n

      a. [latex]\\begin{array}{l}3x<6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)<\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x<2\\hfill \\end{array}[\/latex]\n\n \n\n\n

      b.\u00a0[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill & \\hfill \\\\ -2x\\ge 6\\hfill & \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill & \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n

       <\/p>\n

      c.\u00a0[latex]\\begin{array}{ll}5-x>10\\hfill & \\hfill \\\\ -x>5\\hfill & \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)\\hfill & \\text{Multiply by }-1.\\hfill \\\\ x<-5\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
      \n

      Try It 4<\/h3>\n

      Solve [latex]4x+7\\ge 2x - 3[\/latex].<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n

      Solving Inequalities in One Variable Algebraically<\/h2>\n

      As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.<\/p>\n

      \n

      Example 5: Solving an Inequality Algebraically<\/h3>\n

      Solve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      Solving this inequality is similar to solving an equation up until the last step.<\/p>\n

      [latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill & \\hfill \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill & \\text{Isolate the variable term}.\\hfill \\\\ x\\le 1\\hfill & \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n

      The solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.<\/p>\n<\/div>\n

      \n

      Try It 5<\/h3>\n

      Solve the inequality and write the answer using interval notation: [latex]-x+4<\\frac{1}{2}x+1[\/latex].\n\nSolution<\/a><\/p>\n<\/div>\n

      \n

      Example 6: Solving an Inequality with Fractions<\/h3>\n

      Solve the following inequality and write the answer in interval notation: [latex]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      We begin solving in the same way we do when solving an equation.<\/p>\n

      [latex]\\begin{array}{ll}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill & \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill & \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill & \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill & \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill & \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n

      The solution set is the interval [latex]\\left(-\\infty ,\\frac{15}{34}\\right][\/latex].<\/p>\n<\/div>\n

      \n

      Try It 6<\/h3>\n

      Solve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

      \n\t\t\t

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