Solve an absolute value equation

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 ${8}=\left|{2}x - {6}\right|$ , 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.

⎧ ⎨ ⎩ \begin{matrix} 2 x - 6 = 8 & or & 2 x - 6 = - 8 2 x = 14 & 2 x = - 2 x = 7 & x = - 1 \end{matrix}

$\begin{cases}2x - 6=8\hfill & \text{or}\hfill & 2x - 6=-8\hfill \\ 2x=14\hfill & \hfill & 2x=-2\hfill \\ x=7\hfill & \hfill & x=-1\hfill \end{cases}$

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,

⎧ ⎨ ⎩ \begin{matrix} | x | = 4, | 2 x - 1 | = 3 | 5 x + 2 | - 4 = 9 \end{matrix}

$\begin{cases}|x|=4,\hfill \\ |2x - 1|=3\hfill \\ |5x+2|-4=9\hfill \end{cases}$

A General Note: Solutions to Absolute Value Equations

For real numbers $A$ and $B$ , an equation of the form $|A|=B$ , with $B\ge 0$ , will have solutions when $A=B$ or $A=-B$ . If $B<0$ , the equation $|A|=B$ has no solution.

How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.

Isolate the absolute value term.
Use $|A|=B$ to write $A=B$ or $\mathrm{-A}=B$ , assuming $B>0$ .
Solve for $x$ .

Example 4: Finding the Zeros of an Absolute Value Function

For the function $f\left(x\right)=|4x+1|-7$ , find the values of $x$ such that $\text{ }f\left(x\right)=0$ .

Solution

$\begin{cases}0=|4x+1|-7\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Substitute 0 for }f\left(x\right).\hfill \\ 7=|4x+1|\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Isolate the absolute value on one side of the equation}.\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 7=4x+1\hfill & \text{or}\hfill & \hfill & \hfill & \hfill & -7=4x+1\hfill & \text{Break into two separate equations and solve}.\hfill \\ 6=4x\hfill & \hfill & \hfill & \hfill & \hfill & -8=4x\hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ x=\frac{6}{4}=1.5\hfill & \hfill & \hfill & \hfill & \hfill & \text{ }x=\frac{-8}{4}=-2\hfill & \hfill \end{cases}$

Graph an absolute function with x-intercepts at -2 and 1.5.

Figure 9

The function outputs 0 when $x=1.5$ or $x=-2$ .

Try It 4

For the function $f\left(x\right)=|2x - 1|-3$ , find the values of $x$ such that $f\left(x\right)=0$ .

Solution

Q & A

Should we always expect two answers when solving $|A|=B?$

No. We may find one, two, or even no answers. For example, there is no solution to $2+|3x - 5|=1$ .

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

Isolate the absolute value term.
Use $|A|=B$ to write $A=B$ or $A=\mathrm{-B}$ .
Solve for $x$ .

Example 5: Solving an Absolute Value Equation

Solve $1=4|x - 2|+2$ .

Solution

Isolating the absolute value on one side of the equation gives the following.

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1 = 4 | x - 2 | + 2 - 1 = 4 | x - 2 | - \frac{1}{4} = | x - 2 | \end{matrix}

$\begin{cases}1=4|x - 2|+2\hfill \\ -1=4|x - 2|\hfill \\ -\frac{1}{4}=|x - 2|\hfill \end{cases}$

The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.

Q & A

In Example 5, if $f\left(x\right)=1$ and $g\left(x\right)=4|x - 2|+2$ were graphed on the same set of axes, would the graphs intersect?

No. The graphs of $f$ and $g$ would not intersect. This confirms, graphically, that the equation $1=4|x - 2|+2$ has no solution.

Try It 5

Find where the graph of the function $f\left(x\right)=-|x+2|+3$ intersects the horizontal and vertical axes.

Solution

Absolute Value Functions