Inequalities

When "equal" is not enough

In real life, things are not always exactly equal. A ride might say "you must be at least 120 cm tall". A speed limit says you may drive "up to 30 mph". Your phone needs "more than 10% battery" to update. None of these are single numbers — they are ranges of allowed values. The maths tool for ranges is the inequality.

An inequality compares two quantities that are not (or not only) equal. Instead of =, it uses one of four symbols. Solving inequalities is very close to solving linear equations, with one important new rule you will meet below.

The four symbols

Symbol	Meaning	Example	Example in words
<	less than	x < 5	x is less than 5
>	greater than	x > 5	x is greater than 5
≤	less than or equal to	x ≤ 5	x is 5 or below
≥	greater than or equal to	x ≥ 5	x is 5 or above

A memory trick: the open, wide end of the symbol points to the larger value, and the pointed end points to the smaller one. So in 8 > 3, the wide side faces the 8 (bigger) and the point faces the 3 (smaller). Some people picture the symbol as a hungry mouth that always wants to eat the bigger number.

The line under ≤ and ≥ is the "or equal to" part — it means the boundary number is allowed.

Solving simple inequalities

Here is the good news: you solve an inequality almost exactly like an equation. You use inverse operations and do the same thing to both sides to keep it balanced.

Worked example 1 — one step

Solve x + 4 < 9.

Step 1 — Undo the +4 by subtracting 4 from both sides. x + 4 − 4 < 9 − 4 x < 5

This tells us x can be any number below 5: 4, 2, 0, −1, even 4.999. An inequality usually has infinitely many solutions, which is why we describe them as a range rather than a single value.

Worked example 2 — with division

Solve 3x ≥ 12.

Step 1 — Undo the ×3 by dividing both sides by 3. 3x ÷ 3 ≥ 12 ÷ 3 x ≥ 4

So x is 4 or anything larger. Because we divided by a positive number, the symbol stays the same way round.

Worked example 3 — two steps

Solve 2x − 5 < 7. Just like a two-step equation, undo the −5 first, then the ×2.

Step 1 — Add 5 to both sides. 2x < 12 Step 2 — Divide both sides by 2. x < 6

The one rule that is different: flipping the sign

Here is the only way inequalities behave differently from equations:

When you multiply or divide both sides by a NEGATIVE number, you must FLIP the inequality sign.

Why? Think about a true statement like 3 < 5. Now multiply both sides by −1:

• Left side becomes −3, right side becomes −5.
• Is −3 < −5? No! On a number line, −3 sits to the right of −5, so −3 is actually greater.
• The true statement is −3 > −5.

Multiplying by a negative reversed which number is bigger, so we must reverse the symbol to keep the statement true. This makes sense once you picture the number line: multiplying by a negative reflects every value to the opposite side of zero.

Worked example 4 — the flip in action

Solve −2x < 6.

Step 1 — Divide both sides by −2. Because −2 is negative, flip the < to >. −2x ÷ (−2) > 6 ÷ (−2) x > −3

Check with a value bigger than −3, say x = 0: −2(0) = 0, and 0 < 6 is true. ✓ Now check a value that should fail, x = −4: −2(−4) = 8, and 8 < 6 is false, correctly excluded. The flip gives the right answer.

Note: adding or subtracting a negative does not flip the sign — only multiplying or dividing by a negative does.

Showing solutions on a number line

A number line gives a clear picture of an inequality's solutions. Two things matter — the type of circle and the direction of the arrow:

• Open circle (○) for < or > — the boundary number is not included.
• Closed circle (●) for ≤ or ≥ — the boundary number is included.
• Draw an arrow in the direction of all the values that work.

For x < 5: an open circle on 5, arrow pointing left (toward smaller numbers). For x ≥ 4: a closed circle on 4, arrow pointing right (toward larger numbers).

A quick way to choose the arrow direction: imagine standing on the boundary number. If x is "greater than", walk right; if "less than", walk left.

Worked example 5 — full problem with a number line

Solve 4 − 3x ≤ 13 and describe its number line.

Step 1 — Subtract 4 from both sides. −3x ≤ 9 Step 2 — Divide both sides by −3, and FLIP the sign (dividing by a negative). x ≥ −3

Number line: a closed circle on −3 (because of the "or equal to"), with the arrow pointing right. Every number from −3 upward is a solution.

Practice activity

Turn everyday rules into inequalities, then solve and draw them.

1. Write three real-life limits as inequalities — for example, "I have at most £20 to spend" becomes m ≤ 20, and "the film is for ages 12 and over" becomes a ≥ 12.
2. Solve these and sketch each on a number line, choosing the right open or closed circle: x + 6 > 10, 2x − 1 ≤ 7, and −5x < 20.
3. For the last one, write a sentence explaining why you flipped the sign. Then challenge a partner to spot whether each of your number lines should use an open or closed circle.

Quick recap

Inequalities compare values with <, >, ≤ or ≥ and usually have a whole range of solutions. Solve them like equations with inverse operations, but flip the sign whenever you multiply or divide by a negative. On a number line, use an open circle to exclude the boundary and a closed circle to include it.