How to Solve Linear Inequalities: A Step-by-Step GCSE Algebra Guide

Inequalities look intimidating because of the unfamiliar symbols, but here is the secret every GCSE examiner relies on you eventually working out: solving an inequality is almost identical to solving an equation. If you can rearrange 2x + 3 = 7, you can rearrange 2x + 3 > 7. There is just one extra rule to remember, and one extra step at the end — plotting the answer on a number line.

This guide walks through every type of linear inequality you’ll meet on the Foundation and Higher papers, in the same order a good tutor would teach them.

Treat the inequality sign like an equals sign

Take 2x + 3 > 7. To solve it, do exactly what you’d do with an equation:

1. Subtract 3 from both sides → 2x > 4
2. Divide both sides by 2 → x > 2

That’s the full solution. The value x = 2 is the boundary, and because the sign is “greater than” (not “greater than or equal to”), every number bigger than 2 works.

Tip: Whenever you see <, >, ≤ or ≥, mentally swap it for = while you rearrange. Get the algebra done first, then deal with what the sign actually means at the very end.

Plotting the answer on a number line

A solved inequality isn’t finished until you can represent it. There are two things to decide: the type of circle and the direction of shading.

Symbol	Meaning	Circle	Shade towards
>	greater than	open (hollow)	the right (+∞)
<	less than	open (hollow)	the left (−∞)
≥	greater than or equal to	closed (filled)	the right (+∞)
≤	less than or equal to	closed (filled)	the left (−∞)

So for x > 2, draw an open circle at 2 and shade to the right. The circle is open because 2 itself is not included.

Compare that with ⅓x + 4 ≤ 6. Solving it:

1. Subtract 4 from both sides → ⅓x ≤ 2
2. Multiply both sides by 3 → x ≤ 6

Because the sign is ≤, you draw a closed circle at 6 and shade to the left. The filled circle tells the examiner that 6 is part of the solution.

The one rule that catches everyone out

Here is the single thing that makes inequalities different from equations:

Remember: When you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign.

Look at -3x + 5 > -4. The tempting mistake is to divide by −3 first. Don’t — deal with the +5 first:

1. Subtract 5 from both sides → -3x > -9
2. Divide both sides by −3 → x < 3 ← the sign flips!

-9 ÷ -3 = 3, but because we divided by a negative, the > becomes a <. So you plot a closed circle… no — an open circle at 3 and shade left, giving x < 3. (If the original had been ≥, it would flip to ≤ and the circle would be closed.)

This flip is one of the most common dropped marks on the whole paper. It’s the same kind of small-but-fatal slip students make with sign errors when solving quadratic equations by factorising — getting the method right but losing a mark on a single sign.

Writing the answer in interval notation

Higher tier (and increasingly Foundation) questions want the answer in interval notation. The rule mirrors the circles:

• A closed circle (≤ or ≥) → use a square bracket [ ]
• An open circle (< or >) → use a round bracket / parenthesis ( )
• Always use a parenthesis next to infinity — you can never “reach” infinity.

So our examples become:

Inequality	Number line	Interval notation
x > 2	open circle at 2, shaded right	(2, ∞)
x ≤ 6	closed circle at 6, shaded left	(−∞, 6]
x < 3	open circle at 3, shaded left	(−∞, 3)

”Or” inequalities (two separate solutions)

Sometimes a question gives two inequalities joined by or, such as:

2x − 1 > 7 or −3x + 2 ≥ −1

Solve each one separately:

• 2x − 1 > 7 → 2x > 8 → x > 4 (open circle at 4, shade right)
• −3x + 2 ≥ −1 → −3x ≥ −3 → x ≤ 1 (sign flips; closed circle at 1, shade left)

Both pieces are part of the answer, so in interval notation you join them with a union symbol: (−∞, 1] ∪ (4, ∞).

Compound inequalities (three parts at once)

Finally, you’ll see “sandwich” inequalities like:

−12 < 7x − 5 ≤ 9

The trick is simple: whatever you do, do it to all three parts.

1. Add 5 to all three parts → −7 < 7x ≤ 14
2. Divide all three parts by 7 → −1 < x ≤ 2

So x lies between −1 and 2. On the number line that’s an open circle at −1, a closed circle at 2, with the region between them shaded. In interval notation: (−1, 2].

Did you know? Every operation that’s legal on one side of an inequality is legal on the others — you just have to apply it everywhere at once. That’s why compound inequalities are no harder than the basic ones; there are simply more parts to keep tidy.

Practise until the sign-flip is automatic

Inequalities reward repetition. The algebra is quick once you’ve seen it a few times; the marks are won and lost on the small habits — picking the right circle, flipping the sign at the right moment, and putting brackets in the correct place. These are exactly the kind of low-effort, high-frequency marks we cover in The Non-Calculator Paper: 10 Tactics That Actually Save Marks, and they’re the topics that nudge a borderline grade upward in How to Get a Grade 9 in GCSE Maths.

The fastest way to make the method automatic is to drill varied questions with instant feedback. Our GCSE Algebra app gives you targeted practice on expressions, equations, inequalities, sequences and graphs across both Foundation and Higher tier — so you can hammer inequalities specifically until the sign-flip becomes second nature. If you’d rather revise everything in one place, the GCSE Maths all-in-one app bundles Number, Algebra, Geometry and Statistics together, part of a suite of 2900+ questions across the four subject apps.

Solve it like an equation, flip the sign when you divide by a negative, then plot it carefully. Do that every time and inequalities become some of the most reliable marks on the paper.