{"id":422,"date":"2015-10-26T18:14:54","date_gmt":"2015-10-26T18:14:54","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=422"},"modified":"2015-11-12T18:37:59","modified_gmt":"2015-11-12T18:37:59","slug":"solving-an-absolute-value-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/solving-an-absolute-value-equation\/","title":{"raw":"Solving an Absolute Value Equation","rendered":"Solving an Absolute Value Equation"},"content":{"raw":"Next, we will learn how to solve an absolute value equation<\/strong>. To solve an equation such as [latex]|2x - 6|=8[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is [latex]8[\/latex] or [latex]-8[\/latex]. This leads to two different equations we can solve independently.\r\n
[latex]\\begin{array}{lll}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{array}[\/latex]<\/div>\r\nKnowing how to solve problems involving absolute value functions 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.\r\n
\r\n

A General Note: Absolute Value Equations<\/h3>\r\nThe absolute value of x <\/em>is written as [latex]|x|[\/latex]. It has the following properties:\r\n
[latex]\\begin{array}{l}\\text{If } x\\ge 0,\\text{ then }|x|=x.\\hfill \\\\ \\text{If }x<0,\\text{ then }|x|=-x.\\hfill \\end{array}[\/latex]<\/div>\r\nFor 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.\r\n\r\nAn absolute value equation<\/strong> in the form [latex]|ax+b|=c[\/latex] has the following properties:\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<\/div>\r\n
\r\n

How To: Given an absolute value equation, solve it.<\/h3>\r\n
    \r\n\t
  1. Isolate the absolute value expression on one side of the equal sign.<\/li>\r\n\t
  2. If [latex]c>0[\/latex], write and solve two equations: [latex]ax+b=c[\/latex] and [latex]ax+b=-c[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 8: Solving Absolute Value Equations<\/h3>\r\nSolve the following absolute value equations:\r\n

    a. [latex]|6x+4|=8[\/latex]\r\nb. [latex]|3x+4|=-9[\/latex]\r\nc. [latex]|3x - 5|-4=6[\/latex]\r\nd. [latex]|-5x+10|=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solution<\/h3>\r\na. [latex]|6x+4|=8[\/latex]\r\n\r\nWrite two equations and solve each:\r\n

    [latex]\\begin{array}{ll}6x+4\\hfill&=8\\hfill& 6x+4\\hfill&=-8\\hfill \\\\ 6x\\hfill&=4\\hfill& 6x\\hfill&=-12\\hfill \\\\ x\\hfill&=\\frac{2}{3}\\hfill& x\\hfill&=-2\\hfill \\end{array}[\/latex]<\/p>\r\nThe two solutions are [latex]x=\\frac{2}{3}[\/latex], [latex]x=-2[\/latex].\r\n\r\nb. [latex]|3x+4|=-9[\/latex]\r\n\r\nThere is no solution as an absolute value cannot be negative.\r\n\r\nc. [latex]|3x - 5|-4=6[\/latex]\r\n\r\nIsolate the absolute value expression and then write two equations.\r\n

    [latex]\\begin{array}{lll}\\hfill & |3x - 5|-4=6\\hfill & \\hfill \\\\ \\hfill & |3x - 5|=10\\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill \\\\ 3x - 5=10\\hfill & \\hfill & 3x - 5=-10\\hfill \\\\ 3x=15\\hfill & \\hfill & 3x=-5\\hfill \\\\ x=5\\hfill & \\hfill & x=-\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\r\nThere are two solutions: [latex]x=5[\/latex], [latex]x=-\\frac{5}{3}[\/latex].\r\n\r\nd. [latex]|-5x+10|=0[\/latex]\r\n\r\nThe equation is set equal to zero, so we have to write only one equation.\r\n
    [latex]\\begin{array}{l}-5x+10\\hfill&=0\\hfill \\\\ -5x\\hfill&=-10\\hfill \\\\ x\\hfill&=2\\hfill \\end{array}[\/latex]<\/div>\r\nThere is one solution: [latex]x=2[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Try It 7<\/h3>\r\nSolve the absolute value equation: [latex]|1 - 4x|+8=13[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

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

    [latex]\\begin{array}{lll}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{array}[\/latex]<\/div>\n

    Knowing how to solve problems involving absolute value functions 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

    \n

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

    The absolute value of x <\/em>is written as [latex]|x|[\/latex]. It has the following properties:<\/p>\n

    [latex]\\begin{array}{l}\\text{If } x\\ge 0,\\text{ then }|x|=x.\\hfill \\\\ \\text{If }x<0,\\text{ then }|x|=-x.\\hfill \\end{array}[\/latex]<\/div>\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.\n\nAn absolute value equation<\/strong> in the form [latex]|ax+b|=c[\/latex] has the following properties:<\/p>\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<\/div>\n
    \n

    How To: Given an absolute value equation, solve it.<\/h3>\n
      \n
    1. Isolate the absolute value expression on one side of the equal sign.<\/li>\n
    2. If [latex]c>0[\/latex], write and solve two equations: [latex]ax+b=c[\/latex] and [latex]ax+b=-c[\/latex].<\/li>\n<\/ol>\n<\/div>\n
      \n

      Example 8: Solving Absolute Value Equations<\/h3>\n

      Solve the following absolute value equations:<\/p>\n

      a. [latex]|6x+4|=8[\/latex]
      \nb. [latex]|3x+4|=-9[\/latex]
      \nc. [latex]|3x - 5|-4=6[\/latex]
      \nd. [latex]|-5x+10|=0[\/latex]<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      a. [latex]|6x+4|=8[\/latex]<\/p>\n

      Write two equations and solve each:<\/p>\n

      [latex]\\begin{array}{ll}6x+4\\hfill&=8\\hfill& 6x+4\\hfill&=-8\\hfill \\\\ 6x\\hfill&=4\\hfill& 6x\\hfill&=-12\\hfill \\\\ x\\hfill&=\\frac{2}{3}\\hfill& x\\hfill&=-2\\hfill \\end{array}[\/latex]<\/p>\n

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

      b. [latex]|3x+4|=-9[\/latex]<\/p>\n

      There is no solution as an absolute value cannot be negative.<\/p>\n

      c. [latex]|3x - 5|-4=6[\/latex]<\/p>\n

      Isolate the absolute value expression and then write two equations.<\/p>\n

      [latex]\\begin{array}{lll}\\hfill & |3x - 5|-4=6\\hfill & \\hfill \\\\ \\hfill & |3x - 5|=10\\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill \\\\ 3x - 5=10\\hfill & \\hfill & 3x - 5=-10\\hfill \\\\ 3x=15\\hfill & \\hfill & 3x=-5\\hfill \\\\ x=5\\hfill & \\hfill & x=-\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\n

      There are two solutions: [latex]x=5[\/latex], [latex]x=-\\frac{5}{3}[\/latex].<\/p>\n

      d. [latex]|-5x+10|=0[\/latex]<\/p>\n

      The equation is set equal to zero, so we have to write only one equation.<\/p>\n

      [latex]\\begin{array}{l}-5x+10\\hfill&=0\\hfill \\\\ -5x\\hfill&=-10\\hfill \\\\ x\\hfill&=2\\hfill \\end{array}[\/latex]<\/div>\n

      There is one solution: [latex]x=2[\/latex].<\/p>\n<\/div>\n

      \n

      Try It 7<\/h3>\n

      Solve the absolute value equation: [latex]|1 - 4x|+8=13[\/latex].<\/p>\n

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

      \n\t\t\t

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