{"id":824,"date":"2025-07-15T20:10:11","date_gmt":"2025-07-15T20:10:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=824"},"modified":"2026-01-12T18:27:25","modified_gmt":"2026-01-12T18:27:25","slug":"absolute-value-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions-learn-it-3\/","title":{"raw":"Absolute Value Functions: Learn It 3","rendered":"Absolute Value Functions: Learn It 3"},"content":{"raw":"

Absolute Value Equations<\/h2>\r\nAn absolute value equation<\/strong> is an equation in which the variable of interest is contained within absolute value bars. Absolute value is defined as a distance. That is, the bars are used to designate that the number inside the absolute value represents its distance from zero on the number line.\r\n\r\n
\r\n

absolute value equation<\/h3>\r\nThe general form of an absolute value equation is:\r\n

[latex]|A| = B[\/latex]<\/p>\r\nwhere [latex]A[\/latex] is the expression and [latex]B[\/latex] is a non-negative number.\r\n\r\n \r\n

    \r\n \t
  • 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.<\/li>\r\n<\/ul>\r\n \r\n
      \r\n \t
    • An absolute value equation<\/strong> in the form [latex]|ax+b|=c[\/latex] has the following properties:<\/li>\r\n<\/ul>\r\n
      [latex]\\begin{array}{l}\\text{If }c<0,|ax+b|=c\\text{ has no solution}.\\hfill \\\\ \\text{If }c=0,|ax+b|=c\\text{ has one solution}.\\hfill \\\\ \\text{If }c>0,|ax+b|=c\\text{ has two solutions}.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/section>
      How To: Given an absolute value equation, solve it<\/strong>\r\n
        \r\n \t
      1. Isolate the Absolute Value Expression:<\/strong>\r\nStart by getting the absolute value expression by itself on one side of the equation. This typically involves simplifying the equation and moving any terms that are not inside the absolute value to the other side.<\/li>\r\n \t
      2. Write Two Separate Equations:<\/strong>\r\nSince the absolute value of a number [latex]x[\/latex] an either be [latex]x[\/latex] itself if [latex]x[\/latex] is positive or [latex]-x [\/latex]if [latex]x[\/latex] is negative, split the equation into two cases:\r\n
          \r\n \t
        • Case 1: The expression inside the absolute value equals the value on the other side.<\/li>\r\n \t
        • Case 2: The expression inside the absolute value equals the negative of the value on the other side.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Solve Each Equation Separately:<\/strong>\r\nSolve the two equations that result from step 2. This may involve further simplifications or solving quadratic equations, depending on the form of the original equation.<\/li>\r\n \t
        • Check for Extraneous Solutions:<\/strong>\r\nSince absolute value equations can introduce solutions that do not actually satisfy the original equation when plugged back in, it\u2019s crucial to substitute each found solution back into the original equation to verify its validity.<\/li>\r\n<\/ol>\r\n<\/section>
          The absolute value of both [latex]-3[\/latex] and [latex]3[\/latex] is [latex]3[\/latex] because both are three units away from zero on the number line.\"\"\r\nIn absolute value notation:\r\n

          [latex]|-3| = 3[\/latex] and [latex]|3| = 3[\/latex]\u00a0<\/span><\/p>\r\n\r\n<\/section>

          Solve the absolute value equation:
          [latex]|6x+4|=8[\/latex]<\/center>\r\n\r\n
          \r\n\r\n
          [latex]\\begin{align*} \\text{Original equation} & : & |6x + 4| &= 8 \\\\ \\text{Remove the absolute value, consider both cases} & : & 6x + 4 &= 8 & 6x + 4 &= -8 \\\\ \\text{Subtract 4 from both sides} & : & 6x &= 4 & 6x &= -12 \\\\ \\text{Divide both sides by 6} & : & x &= \\frac{4}{6} & x &= \\frac{-12}{6} \\\\ \\text{Simplify each fraction} & : & x &= \\frac{2}{3} & x &= -2 \\\\ \\end{align*}[\/latex]<\/center>The two solutions are [latex]x=\\dfrac{2}{3}[\/latex], [latex]x=-2[\/latex].\r\n\r\n<\/section>
          Solve the absolute value equations:\r\n
            \r\n \t
          1. [latex]|3x+4|=-9[\/latex]\r\n[reveal-answer q=\"475549\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"475549\"]The equation [latex]|3x+4|=-9[\/latex] involves the absolute value of an expression being equal to a negative number. By definition, the absolute value of any real number or expression is always non-negative (zero or positive). Therefore, it is not possible for the absolute value of an expression to be equal to a negative number.\r\nConsequently, the equation has no solution because there are no values of [latex]x[\/latex] that can satisfy this condition. The absolute value cannot result in a negative output.[\/hidden-answer]<\/li>\r\n \t
          2. [latex]|3x - 5|-4=6[\/latex]\r\n[reveal-answer q=\"266700\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"266700\"][latex]\\begin{align*} \\text{Original equation} & : & |3x - 5| - 4 &= 6 \\\\ \\text{Isolate the absolute value} & : & |3x - 5| &= 10 \\\\ \\text{Consider both cases} & : & \\\\ \\text{Case 1} & : & 3x - 5 &= 10 & \\quad \\text{Case 2} & : & 3x - 5 &= -10 \\\\ \\text{Add 5 to both sides} & : & 3x &= 15 & \\text{Add 5 to both sides} & : & 3x &= -5 \\\\ \\text{Divide by 3} & : & x &= 5 & \\text{Divide by 3} & : & x &= -\\frac{5}{3} \\end{align*}[\/latex][\/hidden-answer]<\/li>\r\n \t
          3. [latex]|-5x+10|=0[\/latex]\r\n[reveal-answer q=\"905868\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"905868\"][latex]\\begin{align*} \\text{Original equation} & : & |-5x + 10| &= 0 \\\\ \\text{Set the expression inside the absolute value to zero} & : & -5x + 10 &= 0 \\\\ \\text{Solve for $x$} & : & -5x &= -10 \\\\ \\text{Divide by -5}& : & x &= 2 \\end{align*}[\/latex][\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section>
            When solving absolute value equations, the absolute value expression must be isolated on one side of the equation\u00a0before<\/strong> setting up the two cases to remove the absolute value bars.\r\n[latex]\\\\[\/latex]\r\nUse the properties of equality to isolate the absolute value expression, but avoid multiplying into or dividing from any expression inside the bars.<\/section>
            [ohm_question hide_question_numbers=1]318770[\/ohm_question]<\/section>
            [ohm_question hide_question_numbers=1]318771[\/ohm_question]<\/section>
            Should 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 one solution to <\/em>[latex]2+|3x - 5|=2[\/latex].<\/p>\r\n\r\n<\/section>

            An absolute value expression is always positive. If your absolute value equations leads to an absolute value expressions being set equal to a negative value, you have\u00a0no solution<\/strong>. Try to catch this case before you split your absolute value equation into two separate equations.Example: [latex]|x+2|=-1[\/latex] has no solution.<\/section> \r\n\r\n<\/section>","rendered":"

            Absolute Value Equations<\/h2>\n

            An absolute value equation<\/strong> is an equation in which the variable of interest is contained within absolute value bars. Absolute value is defined as a distance. That is, the bars are used to designate that the number inside the absolute value represents its distance from zero on the number line.<\/p>\n

            \n

            absolute value equation<\/h3>\n

            The general form of an absolute value equation is:<\/p>\n

            [latex]|A| = B[\/latex]<\/p>\n

            where [latex]A[\/latex] is the expression and [latex]B[\/latex] is a non-negative number.<\/p>\n

             <\/p>\n

              \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.<\/li>\n<\/ul>\n

               <\/p>\n

                \n
              • An absolute value equation<\/strong> in the form [latex]|ax+b|=c[\/latex] has the following properties:<\/li>\n<\/ul>\n
                [latex]\\begin{array}{l}\\text{If }c<0,|ax+b|=c\\text{ has no solution}.\\hfill \\\\ \\text{If }c=0,|ax+b|=c\\text{ has one solution}.\\hfill \\\\ \\text{If }c>0,|ax+b|=c\\text{ has two solutions}.\\hfill \\end{array}[\/latex]<\/div>\n<\/section>\n
                How To: Given an absolute value equation, solve it<\/strong><\/p>\n
                  \n
                1. Isolate the Absolute Value Expression:<\/strong>
                  \nStart by getting the absolute value expression by itself on one side of the equation. This typically involves simplifying the equation and moving any terms that are not inside the absolute value to the other side.<\/li>\n
                2. Write Two Separate Equations:<\/strong>
                  \nSince the absolute value of a number [latex]x[\/latex] an either be [latex]x[\/latex] itself if [latex]x[\/latex] is positive or [latex]-x[\/latex]if [latex]x[\/latex] is negative, split the equation into two cases:<\/p>\n
                    \n
                  • Case 1: The expression inside the absolute value equals the value on the other side.<\/li>\n
                  • Case 2: The expression inside the absolute value equals the negative of the value on the other side.<\/li>\n<\/ul>\n<\/li>\n
                  • Solve Each Equation Separately:<\/strong>
                    \nSolve the two equations that result from step 2. This may involve further simplifications or solving quadratic equations, depending on the form of the original equation.<\/li>\n
                  • Check for Extraneous Solutions:<\/strong>
                    \nSince absolute value equations can introduce solutions that do not actually satisfy the original equation when plugged back in, it\u2019s crucial to substitute each found solution back into the original equation to verify its validity.<\/li>\n<\/ol>\n<\/section>\n
                    The absolute value of both [latex]-3[\/latex] and [latex]3[\/latex] is [latex]3[\/latex] because both are three units away from zero on the number line.\"\"
                    \nIn absolute value notation:<\/p>\n

                    [latex]|-3| = 3[\/latex] and [latex]|3| = 3[\/latex]\u00a0<\/span><\/p>\n<\/section>\n

                    Solve the absolute value equation:<\/p>\n
                    [latex]|6x+4|=8[\/latex]<\/div>\n
                    \n
                    [latex]\\begin{align*} \\text{Original equation} & : & |6x + 4| &= 8 \\\\ \\text{Remove the absolute value, consider both cases} & : & 6x + 4 &= 8 & 6x + 4 &= -8 \\\\ \\text{Subtract 4 from both sides} & : & 6x &= 4 & 6x &= -12 \\\\ \\text{Divide both sides by 6} & : & x &= \\frac{4}{6} & x &= \\frac{-12}{6} \\\\ \\text{Simplify each fraction} & : & x &= \\frac{2}{3} & x &= -2 \\\\ \\end{align*}[\/latex]<\/div>\n

                    The two solutions are [latex]x=\\dfrac{2}{3}[\/latex], [latex]x=-2[\/latex].<\/p>\n<\/section>\n

                    Solve the absolute value equations:<\/p>\n
                      \n
                    1. [latex]|3x+4|=-9[\/latex]\n