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Guide allo studio > MATH 1314: College Algebra

Solving an Absolute Value Equation

Next, we will learn how to solve an absolute value equation. To solve an equation such as 2x6=8|2x - 6|=8, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is 88 or 8-8. This leads to two different equations we can solve independently.
2x6=8 or 2x6=82x=142x=2x=7x=1\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}
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.

A General Note: Absolute Value Equations

The absolute value of x is written as x|x|. It has the following properties:
If x0, then x=x.If x<0, then x=x.\begin{array}{l}\text{If } x\ge 0,\text{ then }|x|=x.\hfill \\ \text{If }x<0,\text{ then }|x|=-x.\hfill \end{array}
For real numbers AA and BB, an equation of the form A=B|A|=B, with B0B\ge 0, will have solutions when A=BA=B or A=BA=-B. If B<0B<0, the equation A=B|A|=B has no solution. An absolute value equation in the form ax+b=c|ax+b|=c has the following properties:
If c<0,ax+b=c has no solution.If c=0,ax+b=c has one solution.If c>0,ax+b=c has two solutions.\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}

How To: Given an absolute value equation, solve it.

  1. Isolate the absolute value expression on one side of the equal sign.
  2. If c>0c>0, write and solve two equations: ax+b=cax+b=c and ax+b=cax+b=-c.

Example 8: Solving Absolute Value Equations

Solve the following absolute value equations:

a. 6x+4=8|6x+4|=8 b. 3x+4=9|3x+4|=-9 c. 3x54=6|3x - 5|-4=6 d. 5x+10=0|-5x+10|=0

Solution

a. 6x+4=8|6x+4|=8 Write two equations and solve each:

6x+4=86x+4=86x=46x=12x=23x=2\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}

The two solutions are x=23x=\frac{2}{3}, x=2x=-2. b. 3x+4=9|3x+4|=-9 There is no solution as an absolute value cannot be negative. c. 3x54=6|3x - 5|-4=6 Isolate the absolute value expression and then write two equations.
3x54=63x5=103x5=103x5=103x=153x=5x=5x=53\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}
There are two solutions: x=5x=5, x=53x=-\frac{5}{3}. d. 5x+10=0|-5x+10|=0 The equation is set equal to zero, so we have to write only one equation.
5x+10=05x=10x=2\begin{array}{l}-5x+10\hfill&=0\hfill \\ -5x\hfill&=-10\hfill \\ x\hfill&=2\hfill \end{array}
There is one solution: x=2x=2.

Try It 7

Solve the absolute value equation: 14x+8=13|1 - 4x|+8=13. Solution

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