Absolute Value Inequalities

Two types of absolute value inequalities

"Less than" type ($|A| \leq c$): The expression inside is between $-c$ and $c$. This gives a compound inequality: $-c \leq A \leq c$. Solution is an interval (connected region).

"Greater than" type ($|A| \geq c$): The expression is either $\leq -c$ or $\geq c$. This gives two separate inequalities: $A \leq -c$ OR $A \geq c$. Solution is a union of two rays.

Step-by-step: Solve $|x - 4| \leq 3$

Step 1 — Identify the type. This is "less than" → compound inequality.

Step 2 — Rewrite without absolute value:

$-3 \leq x - 4 \leq 3$

Step 3 — Solve by adding 4 to all three parts:

$-3 + 4 \leq x \leq 3 + 4$ $1 \leq x \leq 7$

Step 4 — Solution: $[1, 7]$. All values from 1 to 7, inclusive.

Step 5 — Graph: Closed circles at $1$ and $7$, shade the interval between them.

The distance interpretation

$|x - 4| \leq 3$ means "the distance from $x$ to $4$ is at most $3$." On a number line, start at $4$ and go 3 units in each direction: $4 - 3 = 1$ and $4 + 3 = 7$.

"Greater than" example

$|x - 4| > 3$ means "distance from $x$ to $4$ is MORE than $3$": $x < 1$ or $x > 7$. Solution: $(-\infty, 1) \cup (7, \infty)$.

Try it in the visualization

Adjust the center and radius. The number line shows the interval with correct open/closed circles. Toggle between "within distance" and "outside distance" modes. Test points verify which values satisfy the inequality.