Linear Inequalities

≈ 25 min

An inequality compares two expressions: $<, \le, >, \ge$ . Solving works exactly like equations, with one crucial exception:

When you multiply or divide both sides by a negative number, the inequality sign reverses.

Worked example. Solve $2x + 1 < 9$ .

2x < 8 \;\Rightarrow\; x < 4

With the flip. Solve $-3x \ge 12$ .

x \le -4 \qquad \text{(divided by } -3\text{, sign flipped)}

Why? $2 < 5$ is true, but multiply both by $-1$ : $-2 > -5$ . Negation mirrors the number line, so the order swaps.

When does the inequality sign reverse?

Whenever negatives appear anywhere

When adding a negative to both sides

When multiplying or dividing both sides by a negative

Never — inequalities behave exactly like equations

Solve and answer in the form x<… or x>…: $2x + 1 < 9$

Which number line shows $x \le -4$ ?

Open circle at $-4$ , shaded to the right

Filled circle at $-4$ , shaded to the left

Open circle at $-4$ , shaded to the left

Filled circle at 4, shaded to the left

Solve and answer in the form x<=… or x>=…: $-3x \ge 12$