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

What is an inequality?

An inequality compares two expressions with <, >, ≤, or ≥ instead of an equals sign. Its solution is usually a whole range of numbers, not one value.

x < 5 means every number less than 5, such as 4, 0, and −100. x ≤ 5 includes 5 itself. On a number line, an open circle marks a boundary that isn’t included, and a filled circle marks one that is.

How do you solve inequalities step by step?

To solve a linear inequality, isolate x using the same steps you’d use for an equation: clear brackets, combine like terms, and undo addition before multiplication. The only extra rule is about negative numbers, and it gets its own section below.

Solve 3x − 4 < 11. Add 4 to both sides to get 3x < 15. Divide both sides by 3 to get x < 5. Since 3 is positive, the sign stays the same.

When do you flip the inequality sign?

You flip the inequality sign when you multiply or divide both sides by a negative number. That is the only time it flips in a linear inequality.

Here’s why. Start with 2 < 5, which is true. Multiply both sides by −1 and you get −2 and −5. But −2 is bigger than −5, so the true statement is now −2 > −5. Multiplying by a negative reverses the order of numbers on the number line, so the sign has to turn around to stay true.

What does a sign flip look like in a real problem?

Solve −2x + 3 ≥ 11. Subtract 3 from both sides to get −2x ≥ 8. Now divide by −2. Because −2 is negative, ≥ becomes ≤, and the answer is x ≤ −4.

Check with x = −5, which should work: −2(−5) + 3 = 13, and 13 ≥ 11 is true. Try x = 0, which shouldn’t: 3 ≥ 11 is false. Good.

The same rule applies to dividing by a negative written as a fraction. For x/(−2) > 3, multiply both sides by −2 and flip: x < −6.

How do you solve inequalities with x on both sides?

Move the x terms to one side and the numbers to the other, then divide. Watch whether the coefficient you end up dividing by is negative.

Solve 5 − 3x > 2x + 20. Subtract 2x from both sides: 5 − 5x > 20. Subtract 5: −5x > 15. Divide by −5 and flip: x < −3.

You can sidestep the flip by moving x terms toward the side where the coefficient stays positive. Adding 3x to both sides gives 5 > 5x + 20, then −15 > 5x, then −3 > x. That’s the same as x < −3, with no division by a negative.

How do you write and graph the solution?

Draw the boundary on a number line, use an open circle for < or > and a filled circle for ≤ or ≥, then shade the side that works. In interval notation, round brackets mean the end isn’t included and square brackets mean it is.

Here’s a multi-step problem to try it on. Solve 4(x + 1) − 3x ≤ 2(x − 1). The left side simplifies to x + 4 and the right to 2x − 2. Subtract x from both sides to get 4 ≤ x − 2, then add 2 to get 6 ≤ x. That reads more naturally as x ≥ 6.

On the number line, put a filled circle at 6 and shade to the right. In interval notation the answer is [6, ∞). Infinity always gets a round bracket, because it isn’t a number you can reach.

Which steps never flip the sign?

Adding or subtracting any number, positive or negative, never flips the sign. Multiplying or dividing by a positive number never flips it either.

People sometimes flip when the answer is negative, but that isn’t the rule. In 3x < −12, dividing by positive 3 gives x < −4 with no flip. Only the number you divide by matters, not the number on the other side.

How do you solve compound inequalities?

For a compound inequality like −3 < 2x + 1 < 7, do every step to all three parts at once until x is alone in the middle.

Subtract 1 from all three parts: −4 < 2x < 6. Divide all three by 2: −2 < x < 3. The solution is every number between −2 and 3, not including either end.

If you divide by a negative, both signs flip. For −4 ≤ 3 − x < 2, subtract 3 to get −7 ≤ −x < −1, then multiply everything by −1 and flip both signs: 7 ≥ x > 1. Written in the usual order, that’s 1 < x ≤ 7.

Compound inequalities joined by “or” work differently. Solve each part on its own and keep both answers. For example, x < −1 or x > 4 means any number outside that gap, and on a number line you shade two separate pieces.

How do you solve quadratic inequalities?

Find where the quadratic equals zero, mark those roots on a number line, and test one number in each region to see where the inequality holds.

For x² − x − 6 < 0, factor to (x − 3)(x + 2), so the roots are −2 and 3. Test 0, which sits between them: 0 − 0 − 6 = −6, and −6 < 0 is true. Test 4: 16 − 4 − 6 = 6, which isn’t less than 0. The answer is −2 < x < 3.

For x² − 4 ≥ 0, the roots are −2 and 2. Here the outer regions work, so the answer is x ≤ −2 or x ≥ 2. The inequality solver on this site builds this sign chart for you, which is handy for checking.

What mistakes should you watch for when solving inequalities?

The two classics are forgetting to flip when dividing by a negative, and flipping when you didn’t need to. A third is mixing up open and filled circles on the graph. And when you rewrite 6 ≤ x as x ≥ 6, the sign turns around because the sides swapped, not because anything was negative.

The quickest catch is a test value. Pick a number your answer says should work and plug it into the original inequality. If it fails, the sign is pointing the wrong way. The article on checking algebra answers by substitution shows the full three-number test.