In Other Type of Equations, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. 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
8=
2x−6 ,
8=
2x−6 , 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.
2x−6
=
8
or
2x−6
=
−8
2x
=
14
2x
=
−2
x
=
7
x
=
−1
2x−6
=
8
or
2x−6
=
−8
2x
=
14
2x
=
−2
x
=
7
x
=
−1
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.
An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,
 x =4,

2x−1
=3,or

5x+2
−4=9
 x =4,

2x−1
=3,or

5x+2
−4=9
For real numbers
A
A and
B
B, an equation of the form
 A =B,
A=B, with
B≥0,
B≥0, will have solutions when
A=B
A=B or
A=−B.
A=−B. If
B<0,
B<0, the equation
 A =B
A=B has no solution.
Given the formula for an absolute value function, find the horizontal intercepts of its graph.
 Isolate the absolute value term.
 Use
 A =B
 A =B to write
A=B
A=B or
−A=B,
−A=B,
assuming
B>0.
B>0.
 Solve for
x.
x.
For the function
f(x)=4x+1−7,
f(x)=4x+1−7,
find the values of
x
x
such that
f(x)=0.
f(x)=0.
0
=
4x+1−7
Substitute 0 for f(x).
7
=
4x+1
Isolate the absolute value on one side of the equation.
7
=
4x+1
or
−7
=
4x+1
Break into two separate equations and solve.
6
=
4x
−8
=
4x
x
=
6
4
=1.5
x
=
−8
4
=−2
0
=
4x+1−7
Substitute 0 for f(x).
7
=
4x+1
Isolate the absolute value on one side of the equation.
7
=
4x+1
or
−7
=
4x+1
Break into two separate equations and solve.
6
=
4x
−8
=
4x
x
=
6
4
=1.5
x
=
−8
4
=−2
The function outputs 0 when
x=
3
2
x=
3
2
or
x=−2.
x=−2. See Figure 8.
For the function
f(x)=
2x−1
−3,
f(x)=
2x−1
−3,
find the values of
x
x
such that
f(x)=0.
f(x)=0.
Should we always expect two answers when solving
 A =B?
 A =B?
No. We may find one, two, or even no answers. For example, there is no solution to
2+
3x−5 =1.
2+
3x−5 =1.
Analysis
Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.