Inequalities: the signs, number lines and solving them

What is an inequality in algebra?

An inequality in algebra is a statement that one amount is bigger or smaller than another, where a letter stands for an unknown number. An equation like x + 3 = 7 has one answer, x = 4. An inequality like x + 3 > 7 has lots: any number bigger than 4 works, such as 5, 4.5 or 100.

Children first use the < and > signs in Year 2, to compare numbers up to 100, for example 36 < 63. At secondary school the key stage 3 programme of study adds the "or equal to" signs and the word inequality itself. Solving inequalities is in the key stage 4 programme of study, for GCSE.

The four inequality signs

Sign	Means	Example	Numbers that fit
<	less than	x < 3	2, 0, −5, 2.9 (not 3)
>	greater than	x > 3	4, 3.1, 50 (not 3)
≤	less than or equal to	x ≤ 3	3, 2, −1
≥	greater than or equal to	x ≥ 3	3, 4, 10

To remember which is which: the wide end of the sign faces the bigger number, as in 2 < 9. The line underneath in ≤ and ≥ is half of an equals sign, so it means "or equal to".

How to show an inequality on a number line

To show an inequality on a number line, draw a circle at the number, then a line or arrow over all the numbers that fit. The circle tells you whether the number itself is included:

• An open circle (empty in the middle) means the number is not included. Use it for < and >.
• A closed circle (filled in) means the number is included. Use it for ≤ and ≥.

Worked example: x > 2 and x ≤ −1

For x > 2, draw an open circle at 2, because 2 itself doesn't fit (2 isn't greater than 2). Then draw an arrow from the circle to the right, towards the bigger numbers, because 3, 10 and 2.5 all fit.

For x ≤ −1, draw a closed circle at −1, because −1 does fit. Then draw an arrow to the left, towards −2, −3 and smaller. Remember that −3 is smaller than −1: the negative numbers guide explains why.

Worked example: −2 < x ≤ 3

This inequality means x is bigger than −2 and also less than or equal to 3. On the number line it's a line segment between two circles, with no arrows: an open circle at −2 (not included), a closed circle at 3 (included), and a line joining them.

If x has to be a whole number (an integer), the numbers that fit are −1, 0, 1, 2 and 3. −2 isn't included, but 3 is.

Write down the inequality shown on the number line

To write down the inequality shown on a number line, look at the circles first, then the direction:

1. Which number is the circle on? That's the number in your inequality.
2. Open or closed? Open means < or >. Closed means ≤ or ≥.
3. Which way does the arrow go? Right means x is bigger, left means x is smaller.

For example, a closed circle at 4 with an arrow pointing left shows x ≤ 4. An open circle at −3 joined to a closed circle at 1 shows −3 < x ≤ 1.

The same notation turns up with rounding. If a length is 20 cm to the nearest 10 cm, the real length l is in the range 15 ≤ l < 25. The key stage 3 programme of study uses inequalities in exactly this way.

How to solve inequalities

To solve a linear inequality, use the same steps as solving equations: do the same thing to both sides until the letter is on its own. The answer is an inequality too.

Worked example: 3x + 4 > 19

• Take 4 from both sides: 3x > 15, because 19 − 4 = 15.
• Divide both sides by 3: x > 5, because 15 ÷ 3 = 5.

On a number line that's an open circle at 5 with an arrow to the right. Check with a number that should work, like 6: 3 × 6 = 18, and 18 + 4 = 22, which is bigger than 19.

Worked example: 2x − 3 ≤ 7

• Add 3 to both sides: 2x ≤ 10.
• Divide both sides by 2: x ≤ 5.

That's a closed circle at 5, because x = 5 works: 2 × 5 = 10, and 10 − 3 = 7.

The one rule that's different

If you multiply or divide both sides by a negative number, the inequality sign turns round. For example, 2 < 5 is true, but multiply both sides by −1 and you get −2 and −5, and −2 > −5.

So to solve −2x < 6, divide both sides by −2 and turn the sign round: x > −3. Check with x = 0: −2 × 0 is 0, and 0 < 6 is true.

At GCSE, more highly attaining pupils also meet inequalities with two letters, quadratic inequalities and showing inequalities on a graph. The key stage 4 programme of study lists these as extra content.

Common mistakes with inequalities

• Using the wrong circle. Open for < and >, closed for ≤ and ≥. Saying "is the number itself allowed?" sorts it out.
• Pointing the arrow the wrong way. x > 2 points right, towards bigger numbers. Reading the sign out loud ("x is greater than 2") helps.
• Reading an inequality backwards. 5 > x means the same as x < 5.
• Forgetting to turn the sign round when multiplying or dividing by a negative number.
• Giving one answer. x = 6 isn't the solution to 3x + 4 > 19. The solution is x > 5.

How to help at home

• Use everyday limits. A ride that says "you must be at least 120 cm" is h ≥ 120. A sign saying "under 5s go free" is age < 5. Ask your child to write them down.
• Draw the number line. A quick line from −5 to 5 on paper makes most inequality questions easier. The number line guide shows how children use them at primary school.
• Test a number. After solving, pick a number that should fit and one that shouldn't, and check both.
• Ask the school about the order. Key stage 3 covers Years 7 to 9 and schools decide when inequalities are taught, so their scheme of work or curriculum map is the place to look.

For the other topics in these years, see the KS3 maths guide.

Inequalities questions to try

1. Is 7 ≤ 7 true or false?
2. Describe how you would show x < 4 on a number line.
3. Write down the inequality shown by a closed circle at −2 with an arrow pointing right.
4. List the integers that fit −1 ≤ x < 3.
5. Solve 4x + 1 > 21.
6. Solve 5x − 2 ≤ 13.

Answers

1. True: 7 is equal to 7, and ≤ includes "equal to".
2. An open circle at 4 (4 isn't included) with an arrow pointing left.
3. x ≥ −2
4. −1, 0, 1, 2 (−1 is included, 3 isn't)
5. x > 5 (take 1 from both sides to get 4x > 20, then divide by 4)
6. x ≤ 3 (add 2 to both sides to get 5x ≤ 15, then divide by 5)