Absolute Value Equations

Definition of Absolute Value

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

Key properties:

• $|a| \geq 0$ always
• $|a| = 0$ only if $a = 0$
• $|-a| = |a|$

Solving $|ax + b| = c$

If $c > 0$: Split into two cases.

Case 1: $ax + b = c$

Case 2: $ax + b = -c$

Example: $|2x - 3| = 7$

Case 1: $2x - 3 = 7$ → $2x = 10$ → $x = 5$

Case 2: $2x - 3 = -7$ → $2x = -4$ → $x = -2$

Solutions: $x = 5$ or $x = -2$.

If $c = 0$: Only one solution. $ax + b = 0$.

$|3x + 6| = 0$ → $3x + 6 = 0$ → $x = -2$

If $c < 0$: No solution. Absolute value can never be negative.

$|x + 4| = -3$ → no solution.

Isolate the Absolute Value First

If the equation has extra terms, isolate the absolute value before splitting.

Example: $3|x - 1| + 2 = 14$

$3|x - 1| = 12$

$|x - 1| = 4$

Case 1: $x - 1 = 4$ → $x = 5$

Case 2: $x - 1 = -4$ → $x = -3$

Absolute Value with Variables on Both Sides

$|x + 2| = 3x - 4$

Case 1: $x + 2 = 3x - 4$ → $6 = 2x$ → $x = 3$. Check: $|5| = 5$ and $9 - 4 = 5$ ✓

Case 2: $x + 2 = -(3x - 4)$ → $x + 2 = -3x + 4$ → $4x = 2$ → $x = \frac{1}{2}$. Check: $|2.5| = 2.5$ and $1.5 - 4 = -2.5$. $2.5 \neq -2.5$ ✗ Extraneous.

Solution: $x = 3$ only.

Tip 1: Check If the Right Side Is Negative

If $|\text{expression}| = \text{negative}$, immediately write "no solution" and move on.

Tip 2: Always Isolate First

Don't split into cases until the absolute value expression is alone on one side.

Tip 3: Check Both Solutions

Especially when the equation has variables outside the absolute value, one solution may be extraneous.

Tip 4: |a| = |b| Means a = b or a = −b

If you see $|x - 3| = |2x + 1|$, set up: $x - 3 = 2x + 1$ or $x - 3 = -(2x + 1)$.

Tip 5: Distance Interpretation

$|x - 5| = 3$ means "$x$ is 3 units from 5" → $x = 8$ or $x = 2$.