Absolute value equation: |ax+b| = k has 0, 1, or 2 solutions

The rule

For $|A| = k$:

• If $k < 0$: no solution ($|A| \geq 0$ always).
• If $k = 0$: $A = 0$ (1 equation to solve).
• If $k > 0$: $A = k$ or $A = -k$ (2 equations to solve).

Typical example

Solve $|2x - 3| = 5$.

$k = 5 > 0$, two cases:

• $2x - 3 = 5$ ⇒ $x = 4$
• $2x - 3 = -5$ ⇒ $x = -1$

Solutions: $S = \{-1 \, , \, 4\}$.

Geometric interpretation

$|x - a| = k$ means “distance between $x$ and $a$ is $k$”.

$|x - a| = k$ ⇒ $x = a - k$ or $x = a + k$.

Inequality $|A| \leq k$ (with $k > 0$)

$|A| \leq k \iff -k \leq A \leq k$

Ex: $|x - 3| \leq 2$ ⇒ $-2 \leq x - 3 \leq 2$ ⇒ $1 \leq x \leq 5$.

Inequality $|A| \geq k$ (with $k > 0$)

$|A| \geq k \iff A \leq -k \text{ OR } A \geq k$

Ex: $|x + 1| \geq 3$ ⇒ $x + 1 \leq -3$ or $x + 1 \geq 3$ ⇒ $x \leq -4$ or $x \geq 2$.

Equation $|A| = |B|$

$|A| = |B| \iff A = B \text{ OR } A = -B$

Ex: $|x + 1| = |2x - 5|$ ⇒ $x + 1 = 2x - 5$ or $x + 1 = -(2x - 5)$ ⇒ $x = 6$ or $x = 4/3$.

Common mistakes

1. Missing the case $A = -k$ (only solving one).
2. Writing $|A| = k$ with $k < 0$ as having solutions.
3. Confusing $|A|$ and $A$ (absolute value is always non-negative).
4. Writing $|a + b| = |a| + |b|$ (false in general, it’s $\leq$).

?À toi de jouer

Solve |3x + 6| = 9

A.x = 1 or x = −5B.x = 1 onlyC.x = 3 or x = −3D.No solution

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