{"id":1005,"date":"2015-11-12T18:35:32","date_gmt":"2015-11-12T18:35:32","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1005"},"modified":"2017-03-31T21:18:28","modified_gmt":"2017-03-31T21:18:28","slug":"solve-an-absolute-value-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solve-an-absolute-value-equation\/","title":{"raw":"Solve an absolute value equation","rendered":"Solve an absolute value equation"},"content":{"raw":"

Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]{8}=\\left|{2}x - {6}\\right|[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.<\/p>\r\n\r\n

[latex]\\begin{cases}2x - 6=8\\hfill & \\text{or}\\hfill & 2x - 6=-8\\hfill \\\\ 2x=14\\hfill & \\hfill & 2x=-2\\hfill \\\\ x=7\\hfill & \\hfill & x=-1\\hfill \\end{cases}[\/latex]<\/div>\r\n

Knowing how to solve problems involving absolute value functions<\/strong> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\r\n

An absolute value equation<\/strong> is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\r\n\r\n

[latex]\\begin{cases}|x|=4,\\hfill \\\\ |2x - 1|=3\\hfill \\\\ |5x+2|-4=9\\hfill \\end{cases}[\/latex]<\/div>\r\n
\r\n

A General Note: Solutions to Absolute Value Equations<\/h3>\r\n

For real numbers [latex]A[\/latex] and [latex]B[\/latex], an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex], will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]. If [latex]B<0[\/latex], the equation [latex]|A|=B[\/latex] has no solution.<\/p>\r\n\r\n<\/div>\r\n

\r\n

How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.<\/h3>\r\n
  1. Isolate the absolute value term.<\/li>\r\n\t
  2. Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]\\mathrm{-A}=B[\/latex], assuming [latex]B>0[\/latex].<\/li>\r\n\t
  3. Solve for [latex]x[\/latex].<\/li>\r\n<\/ol><\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 4: Finding the Zeros of an Absolute Value Function<\/h3>\r\n

    For the function [latex]f\\left(x\\right)=|4x+1|-7[\/latex] , find the values of\u00a0[latex]x[\/latex] such that\u00a0[latex]\\text{ }f\\left(x\\right)=0[\/latex] .<\/p>\r\n\r\n<\/div>\r\n

    \r\n
    \r\n

    Solution<\/h3>\r\n

    [latex]\\begin{cases}0=|4x+1|-7\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{Substitute 0 for }f\\left(x\\right).\\hfill \\\\ 7=|4x+1|\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{Isolate the absolute value on one side of the equation}.\\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ 7=4x+1\\hfill & \\text{or}\\hfill & \\hfill & \\hfill & \\hfill & -7=4x+1\\hfill & \\text{Break into two separate equations and solve}.\\hfill \\\\ 6=4x\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & -8=4x\\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ x=\\frac{6}{4}=1.5\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{ }x=\\frac{-8}{4}=-2\\hfill & \\hfill \\end{cases}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"GraphFigure 9<\/b>[\/caption]\r\n

    The function outputs 0 when [latex]x=1.5[\/latex] or [latex]x=-2[\/latex].\r\n<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Try It 4<\/h3>\r\n

    For the function [latex]f\\left(x\\right)=|2x - 1|-3[\/latex], find the values of [latex]x[\/latex] such that [latex]f\\left(x\\right)=0[\/latex].<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

    \r\n

    Q & A<\/strong><\/p>\r\nShould we always expect two answers when solving [latex]|A|=B?[\/latex]<\/strong>\r\n

    No. We may find one, two, or even no answers. For example, there is no solution to\u00a0<\/em>[latex]2+|3x - 5|=1[\/latex].<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    How To: Given an absolute value equation, solve it.<\/h3>\r\n
    1. Isolate the absolute value term.<\/li>\r\n\t
    2. Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]A=\\mathrm{-B}[\/latex].<\/li>\r\n\t
    3. Solve for [latex]x[\/latex].<\/li>\r\n<\/ol><\/div>\r\n
      \r\n
      \r\n
      \r\n

      Example 5: Solving an Absolute Value Equation<\/h3>\r\n

      Solve [latex]1=4|x - 2|+2[\/latex].<\/p>\r\n\r\n<\/div>\r\n

      \r\n

      Solution<\/h3>\r\n

      Isolating the absolute value on one side of the equation gives the following.<\/p>\r\n\r\n

      [latex]\\begin{cases}1=4|x - 2|+2\\hfill \\\\ -1=4|x - 2|\\hfill \\\\ -\\frac{1}{4}=|x - 2|\\hfill \\end{cases}[\/latex]<\/div>\r\n

      The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

      \r\n

      Q & A<\/strong><\/p>\r\nIn Example 5, if [latex]f\\left(x\\right)=1[\/latex] and [latex]g\\left(x\\right)=4|x - 2|+2[\/latex] were graphed on the same set of axes, would the graphs intersect?<\/strong>\r\n

      No. The graphs of [latex]f[\/latex] and [latex]g[\/latex] would not intersect. This confirms, graphically, that the equation [latex]1=4|x - 2|+2[\/latex] has no solution.<\/em><\/p>\r\n\r\n<\/div>\r\n

      \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 10<\/b>[\/caption]\r\n\r\n<\/figure>
      \r\n

      Try It 5<\/h3>\r\n

      Find where the graph of the function [latex]f\\left(x\\right)=-|x+2|+3[\/latex] intersects the horizontal and vertical axes.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section>

      ","rendered":"
      \n

      Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]{8}=\\left|{2}x - {6}\\right|[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.<\/p>\n

      [latex]\\begin{cases}2x - 6=8\\hfill & \\text{or}\\hfill & 2x - 6=-8\\hfill \\\\ 2x=14\\hfill & \\hfill & 2x=-2\\hfill \\\\ x=7\\hfill & \\hfill & x=-1\\hfill \\end{cases}[\/latex]<\/div>\n

      Knowing how to solve problems involving absolute value functions<\/strong> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n

      An absolute value equation<\/strong> is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\n

      [latex]\\begin{cases}|x|=4,\\hfill \\\\ |2x - 1|=3\\hfill \\\\ |5x+2|-4=9\\hfill \\end{cases}[\/latex]<\/div>\n
      \n

      A General Note: Solutions to Absolute Value Equations<\/h3>\n

      For real numbers [latex]A[\/latex] and [latex]B[\/latex], an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex], will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]. If [latex]B<0[\/latex], the equation [latex]|A|=B[\/latex] has no solution.<\/p>\n<\/div>\n

      \n

      How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.<\/h3>\n
        \n
      1. Isolate the absolute value term.<\/li>\n
      2. Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]\\mathrm{-A}=B[\/latex], assuming [latex]B>0[\/latex].<\/li>\n
      3. Solve for [latex]x[\/latex].<\/li>\n<\/ol>\n<\/div>\n
        \n
        \n
        \n

        Example 4: Finding the Zeros of an Absolute Value Function<\/h3>\n

        For the function [latex]f\\left(x\\right)=|4x+1|-7[\/latex] , find the values of\u00a0[latex]x[\/latex] such that\u00a0[latex]\\text{ }f\\left(x\\right)=0[\/latex] .<\/p>\n<\/div>\n

        \n
        \n

        Solution<\/h3>\n

        [latex]\\begin{cases}0=|4x+1|-7\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{Substitute 0 for }f\\left(x\\right).\\hfill \\\\ 7=|4x+1|\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{Isolate the absolute value on one side of the equation}.\\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ 7=4x+1\\hfill & \\text{or}\\hfill & \\hfill & \\hfill & \\hfill & -7=4x+1\\hfill & \\text{Break into two separate equations and solve}.\\hfill \\\\ 6=4x\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & -8=4x\\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ x=\\frac{6}{4}=1.5\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{ }x=\\frac{-8}{4}=-2\\hfill & \\hfill \\end{cases}[\/latex]<\/p>\n

        \"Graph<\/p>\n

        Figure 9<\/b><\/p>\n<\/div>\n

        The function outputs 0 when [latex]x=1.5[\/latex] or [latex]x=-2[\/latex].
        \n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n

        Try It 4<\/h3>\n

        For the function [latex]f\\left(x\\right)=|2x - 1|-3[\/latex], find the values of [latex]x[\/latex] such that [latex]f\\left(x\\right)=0[\/latex].<\/p>\n

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

        \n

        Q & A<\/strong><\/p>\n

        Should we always expect two answers when solving [latex]|A|=B?[\/latex]<\/strong><\/p>\n

        No. We may find one, two, or even no answers. For example, there is no solution to\u00a0<\/em>[latex]2+|3x - 5|=1[\/latex].<\/p>\n<\/div>\n

        \n

        How To: Given an absolute value equation, solve it.<\/h3>\n
          \n
        1. Isolate the absolute value term.<\/li>\n
        2. Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]A=\\mathrm{-B}[\/latex].<\/li>\n
        3. Solve for [latex]x[\/latex].<\/li>\n<\/ol>\n<\/div>\n
          \n
          \n
          \n

          Example 5: Solving an Absolute Value Equation<\/h3>\n

          Solve [latex]1=4|x - 2|+2[\/latex].<\/p>\n<\/div>\n

          \n

          Solution<\/h3>\n

          Isolating the absolute value on one side of the equation gives the following.<\/p>\n

          [latex]\\begin{cases}1=4|x - 2|+2\\hfill \\\\ -1=4|x - 2|\\hfill \\\\ -\\frac{1}{4}=|x - 2|\\hfill \\end{cases}[\/latex]<\/div>\n

          The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          \n

          Q & A<\/strong><\/p>\n

          In Example 5, if [latex]f\\left(x\\right)=1[\/latex] and [latex]g\\left(x\\right)=4|x - 2|+2[\/latex] were graphed on the same set of axes, would the graphs intersect?<\/strong><\/p>\n

          No. The graphs of [latex]f[\/latex] and [latex]g[\/latex] would not intersect. This confirms, graphically, that the equation [latex]1=4|x - 2|+2[\/latex] has no solution.<\/em><\/p>\n<\/div>\n

          \n
          \"Graph<\/p>\n

          Figure 10<\/b><\/p>\n<\/div>\n<\/figure>\n

          \n

          Try It 5<\/h3>\n

          Find where the graph of the function [latex]f\\left(x\\right)=-|x+2|+3[\/latex] intersects the horizontal and vertical axes.<\/p>\n

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

          <\/section>\n\n\t\t\t
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