Solving Equations and Inequalities with Absolute Value (Section 3.5)

Equations and Inequalities with Absolute Value

Introduction

This section covers methods for solving equations and inequalities that involve absolute value expressions. Absolute value equations and inequalities are fundamental in algebra and precalculus, as they describe distances and ranges on the real number line.

Equations with Absolute Value

Absolute value equations have the general form $|X| = a$, where $X$ is an algebraic expression and $a$ is a constant.

• Definition: The absolute value of a number $x$, denoted $|x|$, is its distance from zero on the number line, always non-negative.

• General Solution: For $a > 0$ and any algebraic expression $X$:

$|X| = a$ is equivalent to $X = -a$ or $X = a$

• Example: Solve $|x| = 5$

$x = -5$ or $x = 5$

• Both solutions are 5 units from 0 on the number line.

• Example: Solve $|x - 3| - 1 = 4$

First, isolate the absolute value: $|x - 3| = 5$ $x - 3 = -5$ or $x - 3 = 5$ $x = -2$ or $x = 8$

• Check both solutions by substitution to verify they satisfy the original equation.

Special Cases

• If $a = 0$, $|X| = 0$ is equivalent to $X = 0$.

• If $a < 0$, $|X| = a$ has no solution, since absolute value cannot be negative. The solution set is the empty set, denoted $\varnothing$.

Inequalities with Absolute Value

Absolute value inequalities describe ranges of values at a certain distance from a point. The solution method depends on the form of the inequality.

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

Inequality	Equivalent Form
$|X| < a$	$-a < X < a$
$|X| > a$	$X < -a$ or $X > a$
$|X| \leq a$	$-a \leq X \leq a$
$|X| \geq a$	$X \leq -a$ or $X \geq a$

• Example: $|x| < 3$ is equivalent to $-3 < x < 3$

• Example: $|y| \geq 1$ is equivalent to $y \leq -1$ or $y \geq 1$

• Example: $|2x + 3| \leq 4$ is equivalent to $-4 \leq 2x + 3 \leq 4$

Solving and Graphing Solution Sets

• Example: Solve and graph $|3x + 2| < 5$

$-5 < 3x + 2 < 5$ $-7 < 3x < 3$ $-\frac{7}{3} < x < 1$ Solution set: $\left(-\frac{7}{3}, 1\right)$

• Graph the interval on the number line, marking endpoints appropriately.

• Example: Solve and graph $|5 - 2x| \geq 1$

$5 - 2x \leq -1$ or $5 - 2x \geq 1$ $-2x \leq -6$ or $-2x \geq -4$ $x \geq 3$ or $x \leq 2$ Solution set: $(-\infty, 2] \cup [3, \infty)$

• Graph the solution as two intervals on the number line.

Summary Table: Absolute Value Equations and Inequalities

Type	General Form	Solution
Equation	$|X| = a$	$X = a$ or $X = -a$
Inequality (less than)	$|X| < a$	$-a < X < a$
Inequality (greater than)	$|X| > a$	$X < -a$ or $X > a$
Special Case	$|X| = 0$	$X = 0$
No Solution	$|X| = a$ where $a < 0$	Empty set $\varnothing$

Additional info: The notes also emphasize checking solutions by substitution and graphing solution sets on the number line, which is important for visualizing the range of possible values.