Inequalities

Statements comparing two expressions using <, >, ≤, or ≥ instead of =.

Solving 2x <= 8 on the number line

2x <= 8 ⟹ x <= 4

Inequality:

a = 2

b = 8

Note: when a < 0, dividing reverses the inequality direction

Definition

An inequality compares two expressions using one of four symbols:

Solving an inequality means finding all values of the variable that make it true. The answer is usually a range, not a single number.

Solving a simple inequality

Solve $x + 3 < 7$ .

Subtract $3$ from both sides:

$x < 4$

Any number less than $4$ is a solution. On a number line, draw an open circle at $4$ and shade left.

Flip the sign when multiplying or dividing by a negative

If you multiply or divide both sides by a negative number, you must reverse the inequality:

$-2x > 6 \implies x < -3$

This is the one rule that differs from solving equations.

Try it

Solve $3x - 1 \geq 8$ .

Solution

Add $1$ : $3x \geq 9$

Divide by $3$ : $x \geq 3$

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