How to Solve Linear Inequalities (and When to Flip the Sign)

Solve a linear inequality exactly like an equation, with one extra rule: if you multiply or divide both sides by a negative number, flip the inequality sign. The answer is a range of values, which you can draw on a number line or write in interval notation.

The four signs

Worked example 1

Solve $3x - 4 < x + 6$.

1. Subtract $x$ from both sides: $2x - 4 < 6$.
2. Add 4 to both sides: $2x < 10$.
3. Divide by 2 (positive, so no flip): $x < 5$.

In interval notation that is $(-\infty, 5)$.

Worked example 2: dividing by a negative

Solve $-2x + 5 \ge 11$.

1. Subtract 5: $-2x \ge 6$.
2. Divide by $-2$ and flip the sign: $x \le -3$.

Check with a test value: $x = -4$ should work: $-2(-4) + 5 = 13 \ge 11$ ✓. And $x = 0$ should not: $5 \ge 11$ is false ✓.

Special cases

If the $x$ terms cancel, you are left with a statement about numbers:

• $2x + 1 > 2x - 3$ becomes $1 > -3$, which is always true, so every $x$ works.
• $2x + 1 < 2x - 3$ becomes $1 < -3$, which is never true, so there is no solution.

Common mistakes

• Forgetting to flip when dividing by a negative.
• Flipping when subtracting a negative number (that does not flip).
• Using a filled dot for $<$ or $>$.

Practice questions

1. $5x + 2 > 17$
2. $4 - 3x \le 10$
3. $\frac{x}{2} + 1 \le 4$

Answers: 1) $x > 3$ 2) $x \ge -2$ 3) $x \le 6$

Check your answers and see the number line in the inequality solver. For equations with an equals sign, use the linear equation solver, and for absolute values see the absolute value equation solver.

Frequently asked questions

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