Solving Absolute Value Equations

How to solve absolute value equations

The absolute value $|A|$ equals $c$ means $A$ is either $+c$ or $-c$. So $|A| = c$ splits into two separate equations: $A = c$ or $A = -c$. Solve each one independently, then check both answers.

Important: If $c < 0$, there is no solution — absolute value is never negative.

Step-by-step solution: Solve $|2x - 3| = 7$

Step 1 — Check that the right side is non-negative. $7 \geq 0$ ✓, so solutions may exist.

Step 2 — Set up two cases:

Case 1: $2x - 3 = 7$

$2x = 10$ $x = 5$

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

$2x = -4$ $x = -2$

Step 3 — Check both solutions in the original equation.

• $x = 5$: $|2(5) - 3| = |10 - 3| = |7| = 7$ ✓
• $x = -2$: $|2(-2) - 3| = |-4 - 3| = |-7| = 7$ ✓

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

Graphical interpretation

The graph of $y = |2x - 3|$ is a V-shape with vertex at $x = 3/2$. The horizontal line $y = 7$ intersects the V at two points — those x-coordinates are the two solutions.

Common mistakes

• Forgetting the negative case. $|A| = 7$ gives TWO equations, not one.
• Writing $|A| = -5$ and trying to solve. No solution exists — absolute value is always $\geq 0$.
• Not checking for extraneous solutions when the equation is more complex (e.g., $|x - 1| = 2x + 3$ — the negative case can produce extraneous roots).

Try it in the visualization

Adjust $a$, $b$, $c$ in $|ax + b| = c$. The V-shaped graph and horizontal line $y = c$ are drawn — intersections are the solutions. Set $c < 0$ to see "no solution."