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Inequalities

Like an equation but with < or >. Solve the same way, except flip the sign when you multiply or divide by a negative.

What Is an Inequality?

An inequality compares two expressions instead of showing them equal. x > 3 means x can be any number bigger than 3 — not just one value. Solve an inequality the same way you solve an equation: do the same operation to both sides. There is only one extra rule to remember.

Example 1 — easy numbers

Solve x + 3 < 10 . 1 Subtract 3 from both sides: x < 7 Answer: x < 7

Example 2 — dividing by a positive number

Solve 4 x \geq 20 . 1 Divide both sides by 4 (positive, so the sign stays the same): x \geq 5 Answer: x \geq 5

Example 3 — dividing by a negative number

Solve - 3 x > 12 . 1 Divide both sides by - 3 . Because we are dividing by a negative number, flip the sign: x < - 4 Answer: x < - 4

Example 4 — two steps

Solve 2 x - 5 \leq 9 . 1 Add 5 to both sides: 2 x \leq 14 2 Divide both sides by 2 (positive, sign stays the same): x \leq 7 Answer: x \leq 7

Practice

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Solve - 5 x < 20 .

Solve 3 x + 4 \leq 19 . What is the largest whole number x can be?

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