Inequalities — Solving and Representing on a Number Line

An inequality compares two expressions using <, >, ≤ or ≥. Solving an inequality is like solving an equation, except that multiplying or dividing by a negative number reverses the inequality sign.

What is an Inequality?

An inequality compares two expressions. Instead of an equals sign (=), it uses one of:

Symbol	Meaning
`<`	less than
`>`	greater than
≤	less than or equal to
≥	greater than or equal to

Example: $3x + 1 < 10$ means "find all values of $x$ that make $3x+1$ less than 10".

Solving Inequalities

Solve exactly like an equation — except:

Warning

If you multiply or divide both sides by a negative number, you must flip the inequality sign. $-2x > 6 \implies x < -3$ (sign flips because we divided by $-2$ )

Example 1 — Positive coefficient

$3x + 1 < 10$ $3x < 9 \quad (\text{subtract 1 from both sides})$ $x < 3 \quad (\text{divide both sides by } 3)$

Example 2 — Negative coefficient (sign flips)

$5 - 2x \geq 11$ $-2x \geq 6 \quad (\text{subtract 5})$ $x \leq -3 \quad (\text{divide by } -2 \text{, flip sign!})$

Representing on a Number Line

Open circle ○ — the endpoint is not included (use with < or >)
Closed circle ● — the endpoint is included (use with ≤ or ≥)

Inequality	Number line
$x < 3$	open circle at 3, arrow pointing left
$x \geq -1$	closed circle at −1, arrow pointing right

Solution Sets

An inequality has infinitely many solutions — a whole range of values. We can write the solution as:

An inequality: $x < 3$
An interval: $(-\infty, 3)$ — open bracket means not included
A set: $\{x \in \mathbb{R} : x < 3\}$

Real-World Applications

Real world: Height restrictions on rides

Many theme park rides require you to be at least 120 cm tall. If $h$ is your height in cm, the condition is $h \geq 120$ . If you are 115 cm tall, $115 \geq 120$ is false — you cannot ride.

Real world: Speed limits

A speed limit of 50 km/h means every driver's speed $v$ must satisfy $v \leq 50$ . A roadside camera reads 67 km/h — since $67 > 50$ , the driver has exceeded the limit.

Real world: Weight limit in a lift

A lift can hold at most 480 kg. If each person weighs roughly 80 kg and $n$ people enter, the safe condition is $80n \leq 480$ , giving $n \leq 6$ . So at most six people may travel at once.

Tip

Where do you see this? Speed limit signs, age restrictions on apps and websites, weight limits on bridges and lifts, minimum height requirements for theme park rides.

Practice Problems

Solve: 2x − 3 > 7

Show answer

2x > 10, so x > 5. Open circle at 5, arrow pointing right.

Solve: −3x ≤ 12. Which way does the sign flip?

Show answer

Divide both sides by −3 (negative → flip sign): x ≥ −4. Closed circle at −4, arrow pointing right.

A shop says you need to be at least 12 years old to buy a game. Write this as an inequality using the variable a (age).

Show answer

a ≥ 12 — 'at least 12' means 12 is included, so use ≥.