Inequalities, solve them like equations, almost
Learn how to solve inequalities with step-by-step examples and exam-style questions with answers.
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Download the PDF (621 KB)Why does the inequality sign flip when you divide by a negative number?
Inequalities can feel tricky because they’re not just about numbers—they’re about balance. Think of them like a tipped see-saw. When you multiply or divide by a negative, the see-saw tips the other way, so you flip the inequality sign.
The picture to keep in your head: A tipped see-saw
An inequality is a see-saw that's tipped: 2x + 1 > 7 means the left side is heavier. Adding, subtracting, multiplying or dividing both sides by the same positive number keeps it tipped the same way. Multiplying or dividing by a negative number is like turning the see-saw round: the heavy side is now on the other side, so the sign flips. 5 > 2, but -5 < -2.
The method, every time
- Treat the inequality like an equation. Add or subtract the same number from both sides.
- Multiply or divide by a positive number. The see-saw stays tipped the same way.
- Divide by a negative number. Turn the see-saw round and flip the inequality sign.
- Sketch the number line. Use an open circle for < or >, a filled circle for ≤ or ≥.
- Check your answer. Substitute a value from the solution set into the original inequality.
Words you'll see
- inequality
- A mathematical statement showing one value is greater than or less than another.
- flip
- To change the direction of an inequality when multiplying or dividing by a negative.
- interval
- A range of values that satisfy an inequality, shown on a number line.
- quadratic
- An equation with a squared term, often solved by finding where it’s above or below zero.
- boundary
- The exact point where the inequality changes from true to false or vice versa.
Watch it explained
- Why negative times negative is positive (one-minute video)
Same steps as an equation
An inequality is like a tipped see-saw. The left side is heavier than the right side. To solve it, do the same to both sides, like balancing the see-saw. The answer is a range of values, not just one number. This is the same as solving an equation, but the result is a range, not a single value.
Watch out
- Writing x = 3. Use x > 3 to show the range
- Losing the inequality sign halfway. Keep the inequality sign until the end
Divide by a negative? Flip the sign
An inequality is like a tipped see-saw. When you divide by a negative number, the see-saw flips. This means the inequality sign must flip too. If you divide both sides by a negative number, always reverse the sign. You can avoid this by keeping the x terms positive.
Watch out
- Forgetting to flip the sign when dividing by a negative. Always flip the sign when dividing by a negative
- Flipping the sign when adding or subtracting a negative. Only flip when multiplying or dividing by a negative
Open circle, filled circle
An inequality is like a tipped see-saw. When you solve it, you find all the numbers that make it true. On a number line, open circles show numbers that are not included, like < or >. Filled circles show numbers that are included, like ≤ or ≥. For a double inequality, treat all three parts the same. List integer solutions carefully at the ends.
Watch out
- Including an end that isn't included. Check if the inequality is strict (< or >) before listing numbers
- Doing a step to only two of the three parts. Apply the same step to all three parts of the double inequality
Find where the curve is above or below zero
The see-saw works for straight lines; with x2 you need a quick sketch. Solve the equation to find where the U-shaped curve crosses the x-axis: these are the critical values. Between them the curve dips below the axis, outside them it's above. So '< 0' gives one interval between the critical values, and '> 0' gives two parts outside them.
Watch out
- Square-rooting both sides of x2 > 9 and writing only x > 3. Remember to include both x > 3 and x < -3.
- Giving the wrong region (inside instead of outside). Sketch the curve and test values to find where it is below or above zero.
20 exam-style inequalities questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Solve
- 1Solve 2x + 3 > 7[2 marks]
Show the answer
x > 2
2x + 3 > 7 → 2x > 4 → x > 2. The see-saw stays tipped because we subtracted 3 and divided by 2 (positive number). - 2Solve 3x - 4 < 5[2 marks]
Show the answer
x < 3
3x - 4 < 5 → 3x < 9 → x < 3. The see-saw stays tipped because we added 4 and divided by 3 (positive number). - 3Solve 2(x + 1) > 6[2 marks]
Show the answer
x > 2
2(x + 1) > 6 → x + 1 > 3 → x > 2. The see-saw stays tipped because we divided by 2 (positive number). - 4Solve 4x + 1 < 3x + 5[2 marks]
Show the answer
x < 4
4x + 1 < 3x + 5 → x + 1 < 5 → x < 4. The see-saw stays tipped because we subtracted 3x and 1 (positive numbers). - 5Solve (2x + 1)/3 < 3[2 marks]
Show the answer
x < 4
(2x + 1)/3 < 3 → 2x + 1 < 9 → 2x < 8 → x < 4. The see-saw stays tipped because we multiplied by 3 (positive number).
B · Negatives and flips
- 6Solve 7 - 2x > 1[2 marks]
Show the answer
x < 3
Subtract 7: -2x > -6. Divide by -2 and flip: x < 3. - 7Solve -4x ≥ 12[2 marks]
Show the answer
x ≤ -3
Divide by -4 and flip the sign. - 8Solve 3 - x < 8[2 marks]
Show the answer
x > -5
Subtract 3: -x < 5. Multiply by -1 and flip: x > -5. - 9Solve 2x + 9 ≤ 5x without dividing by a negative[2 marks]
Show the answer
x ≥ 3
Subtract 2x from both sides: 9 ≤ 3x, so 3 ≤ x.
C · Integers and double inequalities
- 10List the integers x such that 1 < 2x - 1 ≤ 7.[3 marks]
Show the answer
2, 3, 4
Add 1: 2 < 2x ≤ 8. Divide by 2: 1 < x ≤ 4. - 11Find the largest integer x that satisfies 3x - 2 < 13.[3 marks]
Show the answer
4
3x - 2 < 13. Add 2: 3x < 15. Divide by 3: x < 5. Largest integer less than 5 is 4. - 12Find all integer solutions of 2 < 3x - 4 < 5.[3 marks]
Show the answer
No integer solutions
Add 4: 6 < 3x < 9. Divide by 3: 2 < x < 3. No whole number lies strictly between 2 and 3. - 13Find the smallest integer n such that 4n - 3 > 20.[3 marks]
Show the answer
6
4n > 23, so n > 5.75; the smallest integer is 6.
D · Quadratic (Higher)
- 14Solve x2 < 16[3 marks]
Show the answer
-4 < x < 4
x2 < 16 → x2 - 16 < 0 → (x - 4)(x + 4) < 0. The inequality holds when -4 < x < 4. - 15Solve x2 - 5x + 4 ≥ 0[3 marks]
Show the answer
x ≤ 1 or x ≥ 4
x2 - 5x + 4 = (x - 1)(x - 4). The inequality holds when x ≤ 1 or x ≥ 4. - 16Solve x2 + 2x - 15 > 0[3 marks]
Show the answer
x < -5 or x > 3
x2 + 2x - 15 = (x + 5)(x - 3). The inequality holds when x < -5 or x > 3. - 17A rectangular garden has length (x + 5) m and width (x - 2) m. Find the values of x for which the area is greater than 30 m².[3 marks]
Show the answer
x > 5
(x + 8)(x - 5) > 0 gives x < -8 or x > 5, but the width x - 2 must be positive, so x > 5.
E · Context
- 18A phone plan costs £10 per month plus £0.10 per text. A student has £15 per month. How many texts can they send?[3 marks]
Show the answer
at most 50 texts (x ≤ 50)
Let x = number of texts. 10 + 0.10x ≤ 15. Subtract 10: 0.10x ≤ 5. Divide by 0.10: x ≤ 50. - 19A taxi charges £3 plus £2 per mile. Jo has £20. What is the greatest whole number of miles she can travel?[3 marks]
Show the answer
8 miles
3 + 2m ≤ 20, so 2m ≤ 17 and m ≤ 8.5; the greatest whole number is 8. - 20A bus company charges £2 per ticket. A group of students has £60 to spend. How many tickets can they buy?[3 marks]
Show the answer
at most 30 tickets (x ≤ 30)
Let x = number of tickets. 2x ≤ 60. Divide by 2: x ≤ 30.
Inequalities exam checklist
- Treat it like an equation. Add or subtract the same number from both sides.
- Flip the sign when multiplying or dividing by a negative. The see-saw tips the other way.
- Use an open circle for < or >, a filled circle for ≤ or ≥. Show where the value lies.
- Double inequalities: do all three parts. Solve for each part separately and find the overlap.
- For quadratics: sketch the curve. Find where it’s above or below zero to solve the inequality.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (621 KB)Want maths to click like this?
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