GCSE · Maths · AQA · Spec 8300 · Foundation

Solving linear inequalities in one variable

An equation asks 'which number?'. An inequality asks 'which numbers?', and the answer is usually a whole stretch of the number line.

Try it first

Which numbers make it true?

Find every value of x that makes −4 < 2x ≤ 6 true. Halve each part to get x on its own. Then set both ends of your answer on the line, and decide: does each end number belong in the answer (filled circle) or not (open circle)?

Predict, then check

Picture where these numbers sit on a number line before you choose.

You know 2 < 3. Now multiply both sides by −1. Which statement is true?

Maths · Algebra

Solve it like an equation, with one checkpoint

Two inequalities, same balance method. Step through both and spot the one line where the sign turns round.

GoalFind every x with 3x + 4 < 19
1
3x + 4 < 19
start

Treat it just like the equation 3x + 4 = 19. Your job is to get x on its own, one balanced move at a time.

2
3

Step 1 of 3

Treat it just like the equation 3x + 4 = 19. Your job is to get x on its own, one balanced move at a time.

Watch out: Only multiplying or dividing by a negative number reverses the sign. A negative number that you add or subtract, or a negative answer, changes nothing.

Your turn

Two inequalities in one

Solve −7 ≤ 3x − 1 < 8.

  1. −7 ≤ 3x − 1 < 8This is a combined inequality: 3x − 1 is squeezed between two limits. The easy way in is to treat it as two separate inequalities.
  2. missing step
Which line is step 2?

Spot the slip

Where does this answer go wrong?

Solve 4 − 3x < 19 and describe the answer on a number line.

A student's answer — which line goes wrong?

Strict or included?

Strict: < or >vsIncluded: ≤ or ≥

Every way of writing an answer has to show one thing: is the boundary number in the set or not?

Focus

Does the boundary number work?

Strict: < or >

No. x < 4 means 4 itself is not in the answer.

Included: ≤ or ≥

Yes. x ≤ 4 means 4 is in the answer too.

The insight

The extra line under ≤ is the 'or equal to'. In words: 'more than 4' is x > 4, and 'at most 4' is x ≤ 4.

On a number line

Strict: < or >

Open circle at the boundary

Included: ≤ or ≥

Filled circle at the boundary

In set notation (Higher)

Strict: < or >

{x : x < 4}

Included: ≤ or ≥

{x : x ≤ 4}

On a graph (Higher)

Strict: < or >

Boundary line x = 4 drawn dashed

Included: ≤ or ≥

Boundary line x = 4 drawn solid

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The same balance method you use for equations, with one checkpoint to watch for. And the answer isn't one number. It's a whole stretch of them.

What you need to know

  • The four signs: < less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to.
  • Solve a linear inequality with the balance method: do the same thing to both sides, just as you would for an equation.
  • Multiplying or dividing both sides by a negative number reverses the inequality sign. Adding, subtracting, or multiplying and dividing by a positive number leaves it alone.
  • The answer is a set of values. On a number line, use an open circle for < or > (boundary not included) and a filled circle for ≤ or ≥ (boundary included).
  • A combined inequality like −2 ≤ x < 3 is two inequalities in one. Solve each half, then recombine them.
  • Higher: write the solution set in set notation, such as {x : x > 4}, and on a graph draw the boundary line dashed for < or > and solid for ≤ or ≥.

The big picture

Solve a linear inequality with the balance method, just like an equation. The one checkpoint: multiplying or dividing by a negative number reverses the sign, because it reflects the number line in 0 (2 < 3 becomes −2 > −3). The answer is a set of values. On a number line, an open circle leaves the boundary out and a filled circle includes it. Solve a combined inequality as two separate ones, then recombine. At Higher, you also write the set in set notation and draw graph boundaries dashed (strict) or solid (included).

Key points

1Balance method plus one checkpoint: a negative multiplier or divisor means reverse the sign.
2Why it reverses: multiplying by −1 reflects numbers in 0, so 2 < 3 becomes −2 > −3.
3Open circle = boundary left out. Filled circle = boundary included.
4Check any answer by putting one value from your solution set back into the original inequality.

Worked example

Problem

Solve 3(x − 2) > 5x + 4.

⚠ Watch out

Solving an inequality exactly like an equation, all the way through, and forgetting the checkpoint. Dividing −4x ≤ 12 by −4 gives x ≥ −3, not x ≤ −3. The sign has to reverse whenever you multiply or divide by a negative number.

🧠

Memory hook

Negative? Flip it. Stand a mirror at 0: 2 and 3 swap places with their reflections, so the sign has to swap too.

✓

Check yourself

Solve 9 − 2x ≤ 1, then say how you'd show it on a number line. (Answer: −2x ≤ −8, so x ≥ 4. Filled circle at 4, arrow pointing right.)

Flashcards

(13)
What does x ≤ 7 mean?
x is less than or equal to 7, so 7 itself is included.
When does an inequality sign reverse?
When you multiply or divide both sides by a negative number.
Why does multiplying by −1 reverse an inequality?
It reflects every number in 0 on the number line, which swaps their order. 2 < 3 becomes −2 > −3.
Does subtracting a number reverse the sign?
No. Adding or subtracting never reverses it, even when the number is negative. Only multiplying or dividing by a negative does.
How is the answer to an inequality different from the answer to an equation?
An equation like 2x = 6 has one answer. An inequality has a set of values, shown as a stretch of the number line.
Open circle or filled circle on a number line: what's the difference?
Open circle: the boundary value is left out (< or >). Filled circle: it's included (≤ or ≥).
Which whole numbers satisfy −2 < x ≤ 3?
−1, 0, 1, 2 and 3. −2 is left out (open end); 3 is included (closed end).
How do you solve a combined inequality?
Treat it as two separate inequalities with the same middle, solve each one, then recombine them into one statement.
Why write a combined inequality like −5 < x ≤ 4?
It's more efficient: one statement shows both limits at once.
Solve −5x < 20.
Divide by −5 and reverse the sign: x > −4.
How can you check your answer to an inequality?
Substitute any value from your solution set into the original inequality. It should come out true.
Higher: write x < 4 in set notation.
{x : x < 4}, read as 'the set of all x such that x is less than 4'.
Higher: dashed or solid boundary line on a graph?
Dashed for < or > (points on the line not included). Solid for ≤ or ≥ (points on the line included).

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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