Absolute Value Equations

The Definition

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

Examples

• $|5| = 5$ (5 is already positive)
• $|-5| = 5$ (distance from 0 is 5 units)
• $|0| = 0$ (zero distance)
• $|-3.7| = 3.7$

Key Insight: Absolute value measures how far, not which direction!

Both 5 and -5 are the same distance (5 units) from zero. 📏

The Rule

If $|x| = a$ where $a \geq 0$, then:

$$x = a \text{ or } x = -a$$

Example 1: Solve $|x| = 7$

What numbers are 7 units from zero?

Answer: $x = 7$ or $x = -7$ ✨

Example 2: Solve $|x| = 0$

Only one number is 0 units from zero:

Answer: $x = 0$

Example 3: Solve $|x| = -3$

Distance can't be negative!

No solution ❌

The Strategy

If $|ax + b| = c$ where $c \geq 0$:

1. Set up two equations: $ax + b = c$ and $ax + b = -c$
2. Solve each equation separately
3. Check both solutions in the original equation

Example 1: Solve $|2x - 3| = 5$

Case 1: $2x - 3 = 5$

$2x = 8$ → $x = 4$ ✓

Case 2: $2x - 3 = -5$

$2x = -2$ → $x = -1$ ✓

Solutions: $x = 4$ or $x = -1$

Example 2: Solve $|3x + 1| = 10$

Case 1: $3x + 1 = 10$ → $x = 3$

Case 2: $3x + 1 = -10$ → $x = -\frac{11}{3}$

Less Than ($|x| < a$)

This means "distance from zero is LESS than $a$"

$$|x| < a \text{ means } -a < x < a$$

Example: $|x| < 5$ → $-5 < x < 5$

All numbers between -5 and 5 (not including endpoints)

Greater Than ($|x| > a$)

This means "distance from zero is MORE than $a$"

$$|x| > a \text{ means } x < -a \text{ or } x> a$$

Example: $|x| > 3$ → $x < -3$ or $x> 3$

Numbers less than -3 OR greater than 3

With Expressions

$|x - 2| \leq 3$ means $-3 \leq x - 2 \leq 3$

Add 2 to all parts: $-1 \leq x \leq 5$ ✨

Example: Solve $|x - 1| = |x + 3|$

This means the distance from 1 equals the distance from -3.

Method: Create cases based on the signs

Case 1: Both expressions positive or both negative → same value

$x - 1 = x + 3$ → No solution (impossible!)

Case 2: Opposite signs

$x - 1 = -(x + 3)$

$x - 1 = -x - 3$

$2x = -2$ → $x = -1$ ✓

Solution: $x = -1$

(The point exactly halfway between 1 and -3!)

Watch Out For:

• No Solution: $|x| = -5$ (absolute value can't be negative!)
• One Solution: $|x| = 0$ (only $x = 0$ works)
• All Real Numbers: $|x| \geq 0$ (always true!)
• Empty Set: $|x| < -2$ (impossible)

Common Error

❌ Wrong: Solving $|x + 2| = 5$ as $x + 2 = 5$ only

✅ Right: Need BOTH $x + 2 = 5$ AND $x + 2 = -5$