Algebra and Trigonometry, 5th Edition

3.5 Solving Equations and Inequalities with Absolute Value

• Solve equations with absolute value.

• Solve inequalities with absolute value.

Equations with Absolute Value

JUST IN TIME 3

Recall that the absolute value of a number is its distance from 0 on the number line. We use this concept to solve equations with absolute value.

For $a>0$ and an algebraic expression X:

$$\left|X\right|=a\text{is equivalent to}X=-a\text{or}X=a.$$

Example 1

Solve: $\left|x\right|=5$.

Algebraic Solution

We have

$$\begin{array}{l}\left|x\right|=5\\ x=\mathrm{-5}\mathit{\text{ or }}x=5.& \mathbf{\text{Writing an equivalent}}\hfill \\ \mathbf{\text{statement}}\hfill \end{array}$$

The solutions are −5 and 5.

To check, note that −5 and 5 are both 5 units from 0 on the number line.

Visualizing the Solution

The first ...