Topic 9 Absolute Value Equations

9.1 The Direction of a Number

Can you determine the value of the expression $\frac{| x |}{x}$ for all nonzero real number $x$ and explain the meaning of the value?

9.2 Properties of Absolute Values

The absolute value of a real number $a$ , denoted by $| a |$ , is the distance from $0$ to $a$ on the number line. In particular, $| a |$ is always greater than or equal to $0$ , that is $| a | \geq 0$ . Absolute values satisfy the following properties: $| - a | = | a |, | a b | = | a | | b | and | \frac{a}{b} | = \frac{| a |}{| b |} .$

An absolute value equation may be rewritten as $| X | = c$ , where $X$ represents an algebraic expression.

If $c$ is positive, then the equation $| X | = c$ is equivalent to { $X = c$ or $X = - c$ .}

If $c$ is negative, then the solution set of $| X | = c$ is empty. An empty set is denoted by $\emptyset$ .

More generally, $| X | = | Y |$ is equivalent to $X = Y$ or $X = - Y$ .

The equation $| X | = 0$ is equivalent to $X = 0$ .

Example 9.1 Solve the equation $| 2 x - 3 | = 7.$

Solution.

The equation is equivalent to $\begin{aligned} 2 x - 3 = - 7 & or & 2 x - 3 = 7 \\ 2 x = - 4 & 2 x = 10 \\ x = - 2 & or & x = 5 \end{aligned}$

The solutions are $x = - 2$ or $x = 5$ . In set-builder notation, the solution set is ${- 2, 5}$ .

Example 9.2 Solve the equation $| 2 x - 1 | - 3 = 8.$

Solution.

Rewrite the equation into $| X | = c$ form. $| 2 x - 1 | = 11$
Solve the equation. $\begin{aligned} 2 x - 1 = - 11 & or & 2 x - 1 = 11 \\ 2 x = - 10 & 2 x = 12 \\ x = - 5 & or & x = 6 \end{aligned}$

The solutions are $x = - 5$ or $x = 6$ . In set-builder notation, the solution set is ${- 5, 6}$ .

Example 9.3 Solve the equation $3 | 2 x - 5 | = 9.$

Solution.

Rewrite the equation into $| X | = c$ form. $| 2 x - 5 | = 3$
Solve the equation. $\begin{aligned} 2 x - 5 = - 3 & or & 2 x - 5 = 3 \\ 2 x = 2 & 2 x = 8 \\ x = 1 & or & x = 4 \end{aligned}$

The solutions are $x = 1$ or $x = 4$ . In set-builder notation, the solution set is ${1, 4}$ .

Example 9.4 Solve the equation $2 | 1 - 2 x | - 3 = 7.$

Solution.

Rewrite the equation into $| X | = c$ form. $| 2 x - 1 | = 5$
Solve the equation. $\begin{aligned} 2 x - 1 = - 5 & or & 2 x - 1 = 5 \\ 2 x = - 4 & 2 x = 6 \\ x = - 2 & or & x = 3 \end{aligned}$

The solutions are $x = - 2$ or $x = 3$ . In set-builder notation, the solution set is ${- 2, 3}$ .

Example 9.5 Solve the equation $| 3 x - 2 | = | x + 2 | .$

Solution.

Note that two numbers have the same absolute value only if they are the same or opposite to each other. Then the equation is equivalent to
$3 x - 2 = x + 2 or 3 x - 2 = - (x + 2) .$ $\begin{aligned} 3 x - 2 = x + 2 & or & 3 x - 2 = - (x + 2) \\ 2 x = 4 & 4 x = 0 \\ x = 2 & or & x = 0 \end{aligned}$

The solutions are $x = 2$ and $x = 0$ . In set-builder notation, the solution set is ${0, 2}$ .

Example 9.6 Solve the equation $2 | 1 - x | = | 2 x + 10 | .$

Solution.

Since $2$ is positive, $2 | 1 - x | = | 2 | | 1 - x | = | 2 - 2 x |$ . Moreover, because $| - X | = | X |$ , the equation is equivalent to $| 2 x - 2 | = | 2 x + 10 | .$ $\begin{aligned} 2 x - 2 = 2 x + 10 & or & 2 x - 2 = - (2 x + 10) \\ - 2 = 10 & or & 4 x = - 8 \\ x = - 2 \end{aligned}$

The original equation only has one solution $x = - 2$ . In set-builder notation, the solution set is ${- 2}$ .

9.3 Practice

Problem 9.1 Find the solution set for the equation.

$| 2 x - 1 | = 5$
$| \frac{3 x - 9}{2} | = 3$

Problem 9.2 Find the solution set for the equation.

$| 3 x - 6 | + 4 = 13$
$3 | 2 x - 5 | = 9$

Problem 9.3 Find the solution set for the equation.

$| 5 x - 10 | + 6 = 6$
$- 3 | 3 x - 11 | = 5$

Problem 9.4 Find the solution set for the equation.

$3 | 5 x - 2 | - 4 = 8$
$- 2 | 3 x + 1 | + 5 = - 3$

Problem 9.5 Find the solution set for the equation.

$| 5 x - 12 | = | 3 x - 4 |$
$| x - 1 | = - 5 | (2 - x) - 1 |$

Problem 9.6 Find the solution set for the equation.

$| 2 x - 1 | = 5 - x$
$- 2 x = | x + 3 |$