Two-Step Inequalities

Solve it like an equation

Treat the inequality sign like an equals sign and do your normal solving: move the constant, then divide by the coefficient. For $3x + 4 > 19$, subtract $4$ to get $3x > 15$, then divide by $3$ to get $x > 5$. Adding and subtracting never flip the sign. Multiplying or dividing by a positive number never flips it either.

The answer $x > 5$ means every number bigger than $5$ works: $6$, $10$, $5.001$, all of them. That's why we often graph the solution on a number line — a ray shows the whole range at once. Here is $x > 5$: an open circle at $5$ (because $5$ itself is not included) with the ray running right through every larger number.

When (and why) the sign flips

The flip rule: whenever you multiply or divide both sides by a negative number, reverse the inequality sign. $<$ becomes $>$, and $\le$ becomes $\ge$.

Here's why it makes sense. Start with something true: $2 < 5$. Now multiply both sides by $-1$: you get $-2$ and $-5$. But $-2$ is greater than $-5$ — the negatives reversed the order. Multiplying by a negative mirrors every number across zero, so whichever side was bigger becomes smaller. The flip keeps the statement true.

One warning: the flip is only about multiplying or dividing by a negative. Subtracting a number, or moving a negative term across, does not flip anything. In $x - 7 < 2$, adding $7$ gives $x < 9$ — same sign, no flip.

Check your answer with a test value

Because the solution is a range, you check it by picking an easy number from your answer and plugging it into the original inequality. If you solved $-2x + 7 \le 19$ and got $x \ge -6$, test $x = 0$: the original becomes $7 \le 19$, which is true. If your test value fails, you almost certainly missed a flip.

For an even stronger check, also test a number outside your range and make sure it fails. Ten seconds of testing catches nearly every sign mistake on this topic.

Worked examples

Try one yourself

Common questions