Inequalities

What's different from equations?

Solving inequalities is almost identical to solving equations, with one critical exception: if you multiply or divide both sides by a negative number, flip the inequality sign.

Example. Solve $-3x + 4 > 13$.

1. Subtract 4: $-3x > 9$
2. Divide by $-3$ — and flip the sign: $x < -3$

Solution: any $x$ less than $-3$.

Interval notation

Set	Interval	Meaning
$x > 2$	$(2, \infty)$	open at 2; everything bigger
$x \geq 2$	$[2, \infty)$	closed at 2 (2 is included)
$-1 \leq x < 4$	$[-1, 4)$	closed at $-1$, open at $4$
$x < 0$ or $x > 5$	$(-\infty, 0) \cup (5, \infty)$	union of two intervals

Round brackets $(\,)$ are open — the endpoint is not included. Square brackets $[\,]$ are closed — endpoint is included. Infinity is always open.

Graphing on a number line

• Open dot for $<$ or $>$ (endpoint not included).
• Closed dot for $\leq$ or $\geq$ (endpoint included).
• Shade in the direction the solution extends.

Compound inequalities

"And" (intersection). $2 < x \leq 7$ means $x > 2$ AND $x \leq 7$. Interval: $(2, 7]$.

"Or" (union). $x < -1$ or $x \geq 3$. Interval: $(-\infty, -1) \cup [3, \infty)$.

Another worked example

Solve $5 - 2x \leq 11$.

1. Subtract 5: $-2x \leq 6$
2. Divide by $-2$ and flip: $x \geq -3$

Solution: $[-3, \infty)$.

Key takeaways

• Solve like an equation, but flip the sign when multiplying or dividing by a negative.
• $($ open, $[$ closed, $\infty$ always open.
• Open dot = strict inequality; closed dot = endpoint included.