Algebra · GCSE Maths
Inequalities on a line
GCSE Maths inequalities: solve linear inequalities with the balance method, reverse the sign when multiplying or dividing by a negative, and show the answer on a number line.
Inequalities use the same balance as equations, with one extra rule: multiply or divide by a negative and the sign flips. An open circle is < or >; a filled circle is ≤ or ≥.
The important bits
What you need to know
- 1
Solve 3x + 2 < 11 as you would an equation: subtract 2, 3x < 9, divide by 3, x < 3. Each line must stay a true inequality.
- 2
Multiplying or dividing both sides by a negative reverses the inequality. From −2x > 10, divide by −2 to get x < −5, not x > −5.
- 3
On a number line, < and > use an open circle (the endpoint is not included). ≤ and ≥ use a filled circle. Shade the side that matches the inequality.
- 4
Integer solutions of x < 3 are …, 0, 1, 2 if x is an integer, not 3. Integer solutions of x ≤ 3 include 3. Read “integer” in the question before you list.
- 5
Two inequalities at once describe a segment. 1 < x ≤ 4 is an open circle at 1, filled circle at 4, and the line between them. Write it as a single compound statement if asked.
- 6
Expand brackets and collect like terms before you move x. 2(x − 3) ≥ 5x + 1 becomes 2x − 6 ≥ 5x + 1, then −6 − 1 ≥ 5x − 2x, then −7 ≥ 3x, then x ≤ −7/3.
- 7
Quadratic inequalities (Higher): solve the equation to find the critical values, sketch the U or n, and read where the curve is above or below the axis. Do not treat x² > 9 as x > 3 only.
- 8
Set notation and number-line diagrams are interchangeable. x > 2 is the same as an open circle at 2 with an arrow to the right. Use the form the answer line shows.
Quotations worth analysing
Short evidence. Real method.
“Reverse the inequality when multiplying or dividing by a negative”
This is the extra rule that equations do not have. Leaving the sign unflipped after dividing by −2 is the standard lost mark.
“Open circle for < or > ; closed circle for ≤ or ≥”
The circle is a mark in its own right. A filled circle on a strict inequality is a wrong answer even if the algebra was perfect.
“x < 3”
If the unknown is a real number, the answer is an inequality, not “2”. List integers only when the question says integer values.
Go deeper
Balance, then decide whether the sign flips
Treat the inequality as a balance that still has to stay true. Adding or subtracting the same number on both sides never flips the sign: 5 < 8 remains true if you subtract 10, −5 < −2. Multiplying by a positive number never flips: 2 × 5 < 2 × 8. Multiplying by a negative reverses the comparison: −2 × 5 > −2 × 8, because −10 is greater than −16. That is why −2x > 10 becomes x < −5. If you dislike the flip, add 2x first so the x term is positive: from −2x > 10, add 2x to get 0 > 10 + 2x, then subtract 10, −10 > 2x, then divide by 2 (positive, no flip), −5 > x, which is x < −5. Same answer, no negative division. Use whichever story you will not muddle in the exam.
Go deeper
Number lines are part of the working
Sketch the line, mark the critical value, choose open or filled, then shade. For x ≥ −1, filled circle at −1, arrow right. For −2 < x ≤ 3, open at −2, filled at 3, shade between. Students lose a mark by shading the wrong side after a correct x < 3, usually because they think “less than” means left of 0 rather than left of 3. Test a number: is 0 less than 3? Yes, so 0 must sit in the shaded region. Integer questions: x < 3 and x is an integer means x ≤ 2 if you are listing, so 2 is the largest listed value. Writing x ≤ 2 as an inequality is then equivalent for integers, but it is not equivalent for real x, so only convert when the question restricted to integers. Read the command word: “show on a number line” versus “list the integer values”.
Go deeper
Quadratic inequalities need a sketch, not a slogan
x² > 9 is not just x > 3. The U-shaped graph of y = x² − 9 is positive outside the roots, so x < −3 or x > 3. x² ≤ 9 is the inside: −3 ≤ x ≤ 3. Factor, find the roots, sketch, pick a test point in each region. For (x − 1)(x + 4) ≤ 0 the roots are 1 and −4; the product is negative or zero between them, so −4 ≤ x ≤ 1. Writing x ≤ 1 alone includes x = −10, and (−10 − 1)(−10 + 4) is positive, which does not satisfy ≤ 0. The sketch is not decoration; it is the method. If a is negative, the n-shape flips which region is above the axis. Always test one number from each interval rather than guessing from the inequality sign.
See the idea in action
Solve −2x + 3 ≤ 11 and show the answer on a number line. Then list the integer solutions of −1 < x ≤ 2. Inequality: Step 1: Subtract 3 from both sides: −2x ≤ 8. Step 2: Divide both sides by −2 and reverse the sign: x ≥ −4. Step 3: Number line: filled circle at −4, shade to the right. Check: x = −4 gives −2(−4) + 3 = 11, which is allowed by ≤. x = −5 gives 13, which is not ≤ 11. Integers: Step 4: −1 < x ≤ 2 and x integer means x = 0, 1, 2. Open at −1 so −1 is not included; filled at 2 so 2 is included.
Exam technique
Turn knowledge into marks
If you divide by a negative, write “reverse inequality” in the margin. On the number line, check one test number so the shading cannot wander to the wrong side.
Common mistakes
Do not give these marks away
- 01
Forgetting to reverse the inequality after dividing by a negative coefficient of x.
- 02
Drawing a filled circle on a strict inequality, or shading the wrong side of a correct critical value.
- 03
Solving x² > 9 as only x > 3, missing the branch x < −3.
The solution of −2x > 10 is
Ax < −5
Bx > −5
Cx < 5
Dx > 5
Show the answer
x < −5. Divide by −2 and reverse: x < −5. x > −5 is the unflipped error. x < 5 and x > 5 dropped the sign of 10 as well.
Quick questions
If this is the bit you searched
When do I reverse an inequality?
When you multiply or divide both sides by a negative number. Adding, subtracting, or multiplying by a positive number leaves the sign alone.
How do I show an inequality on a number line?
Open circle for < or >, filled circle for ≤ or ≥, then shade in the direction that makes the inequality true. Test a number if you are unsure which way to shade.
What is the difference between x < 3 and x ≤ 3?
x < 3 does not include 3; x ≤ 3 does. On a number line that is open versus filled. If x must be an integer, x < 3 means 2, 1, 0, …
How do you solve a quadratic inequality?
Rearrange to f(x) > 0 or f(x) < 0, find the roots of f(x) = 0, sketch the parabola, and read the intervals where the sketch is above or below the x-axis.