Compound and Absolute Value Inequalities

Compound Inequalities

"And" Inequalities (Intersection)

Both conditions must be true: $a < x < b$

$-3 < 2x + 1 < 7$ $-4 < 2x < 6$ $-2 < x < 3$

Graph: Open circles at -2 and 3, shade between.

"Or" Inequalities (Union)

At least one condition must be true:

$x < -2 \quad \text{or} \quad x > 5$

Graph: Open circles at -2 and 5, shade left of -2 and right of 5.

Absolute Value Equations

$|ax + b| = c \quad (c \geq 0)$

Split into two equations: $ax + b = c \quad \text{or} \quad ax + b = -c$

Example: $|2x - 3| = 7$ $2x - 3 = 7 \implies x = 5$ $2x - 3 = -7 \implies x = -2$

Absolute Value Inequalities

Less Than: $|x| < c$ → "And"

$|ax + b| < c \implies -c < ax + b < c$

Greater Than: $|x| > c$ → "Or"

$|ax + b| > c \implies ax + b > c \quad \text{or} \quad ax + b < -c$

Example: $|x - 4| \leq 3$ $-3 \leq x - 4 \leq 3$ $1 \leq x \leq 7$

Example: $|2x + 1| > 5$ $2x + 1 > 5 \implies x > 2$ $2x + 1 < -5 \implies x < -3$

Memory trick: "Less thAND" and "GreatOR."