Linear Inequalities

A linear inequality uses $<$, $\leq$, $>$, or $\geq$ instead of $=$. The solution isn't a single value — it's a range of values. For example, $x > 3$ means every number greater than 3.

Most solving moves work exactly like equations:

• Add or subtract any quantity to both sides — sign stays the same.
• Multiply or divide by a positive number — sign stays the same.
• Multiply or divide by a negative number — flip the inequality sign.

That last rule is the only thing that makes inequalities different from equations.

Why does the sign flip? If $5 > 3$, multiply both by $-1$: you get $-5$ on the left, $-3$ on the right. Is $-5 > -3$? No — $-5$ is less than $-3$. The order reversed. Hence the flip.

Compound inequalities — sometimes you'll see $-3 < 2x + 1 \leq 7$. Treat it as two inequalities and do the same operation to all three parts:

1. Subtract 1: $-4 < 2x \leq 6$.
2. Divide by 2: $-2 < x \leq 3$.

The answer is the range $-2 < x \leq 3$ — every value between them, including 3 but not -2.

Common SAT tasks:

• Solve for $x$ and pick the choice with the right range.
• Pick the value that is (or is not) a solution. Plug each option into the original.
• Word problems: translate at least → $\geq$, no more than → $\leq$, fewer than → $<$.

Word translations to memorize:

• At least, no less than, minimum → $\geq$
• At most, no more than, maximum → $\leq$
• Fewer than, less than (strict) → $<$
• More than, greater than (strict) → $>$