Inequalities — Mathematics, 11–13 years

An inequality compares amounts when they are not necessarily equal. It helps describe limits, ranges and choices such as a bag weighing at most 7 kilograms or a game score above 100.

A comparison with room for many answers

The symbols < and > compare numbers: 4 < 9 means 4 is less than 9, while 9 > 4 means 9 is greater than 4. An inequality can be true for many values, not just one. For example, x < 5 includes 4, 0 and -3, but not 5.

Why equality is not enough

Many real questions ask for a safe limit rather than one exact answer: how many people may enter, how old someone must be, or how much weight a shelf can hold. The equals sign alone cannot express “no more than” or “at least”. Inequalities were developed to describe these useful boundaries.

Solving 3x + 2 ≤ 14

First remove 2 from both sides: 3x + 2 ≤ 14 becomes 3x ≤ 12. Then divide both sides by 3: x ≤ 4. So every number up to and including 4 works. Check x = 4: 3 × 4 + 2 = 14, which is allowed because the sign includes equality.

Reading the sign backwards

The pointed end of < faces the smaller number, and its open side faces the larger one. People often reverse it because the symbol looks like an arrow and arrows are usually read from left to right. This is reasonable, so compare the two values aloud before deciding which way the sign points.

Limits in everyday decisions

Inequalities appear on lift signs, medicine instructions, travel rules and product labels. “Maximum load 300 kg” means the load must be less than or equal to 300 kg, while “keep below 5 °C” means the temperature must be less than 5 °C. These statements protect people by setting boundaries.

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