Next, we will learn how to solve an absolute value equation. 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.
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 [latex]|x|[/latex]. It has the following properties:
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.
An absolute value equation in the form [latex]|ax+b|=c[/latex] has the following properties:
How To: Given an absolute value equation, solve it.
- Isolate the absolute value expression on one side of the equal sign.
- If [latex]c>0[/latex], write and solve two equations: [latex]ax+b=c[/latex] and [latex]ax+b=-c[/latex].
Example 8: Solving Absolute Value Equations
Solve the following absolute value equations:
a. [latex]|6x+4|=8[/latex]
b. [latex]|3x+4|=-9[/latex]
c. [latex]|3x - 5|-4=6[/latex]
d. [latex]|-5x+10|=0[/latex]
Solution
a. [latex]|6x+4|=8[/latex]
Write two equations and solve each:
[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]
The two solutions are [latex]x=\frac{2}{3}[/latex], [latex]x=-2[/latex].
b. [latex]|3x+4|=-9[/latex]
There is no solution as an absolute value cannot be negative.
c. [latex]|3x - 5|-4=6[/latex]
Isolate the absolute value expression and then write two equations.
There are two solutions: [latex]x=5[/latex], [latex]x=-\frac{5}{3}[/latex].
d. [latex]|-5x+10|=0[/latex]
The equation is set equal to zero, so we have to write only one equation.
There is one solution: [latex]x=2[/latex].