Inequalities, explained with examples
An inequality compares two values that are not necessarily equal. Solving one is almost the same as solving an equation, with one important extra rule, and the answer is a whole range of values rather than a single number.
On this page
The symbols
- < less than; > greater than.
- ≤ less than or equal to; ≥ greater than or equal to.
x > 3 means x can be any number bigger than 3, such as 3.1, 7 or 1000, but not 3 itself. x ≥ 3 includes 3. A helpful memory aid: the symbol's wide open end faces the bigger value.
Number lines
Inequalities are shown on number lines with circles and arrows. An open circle means the value is not included (< or >). A filled (solid) circle means it is included (≤ or ≥). The arrow or line shows all the values that satisfy the inequality. For −2 < x ≤ 4, draw an open circle at −2, a filled circle at 4, and a line joining them.
Solving linear inequalities
Solve an inequality like an equation: do the same to both sides. You can add or subtract any number, and multiply or divide by a positive number, without changing the sign. But there is one key rule:
When you multiply or divide both sides by a negative number, reverse the inequality sign.
Why? Start with 2 < 5, which is true. Multiply both sides by −1: −2 and −5. Now −2 is greater than −5, so the sign must flip to keep the statement true. A safe alternative is to rearrange so that the x term ends up positive, avoiding division by a negative altogether.
Worked examples
Example 1: a basic inequality
Solve 3x + 5 < 20.
- Subtract 5: 3x < 15. Divide by 3: x < 5.
Example 2: unknowns on both sides
Solve 7x − 4 ≥ 2x + 11.
- Subtract 2x: 5x − 4 ≥ 11. Add 4: 5x ≥ 15.
- x ≥ 3.
Example 3: dividing by a negative
Solve 8 − 2x > 14.
- Subtract 8: −2x > 6. Divide by −2 and flip the sign.
- x < −3. Check: x = −4 gives 8 + 8 = 16, which is greater than 14.
Example 4: a double inequality
Solve −5 ≤ 2x + 1 < 9.
- Do the same to all three parts. Subtract 1: −6 ≤ 2x < 8.
- Divide by 2: −3 ≤ x < 4.
Example 5: integer solutions
List the integers n that satisfy −3 ≤ n < 4.
- Include −3 (because of ≤) but not 4 (because of <).
- −3, −2, −1, 0, 1, 2, 3. Zero is an integer, so do not forget it.
Example 6: a word problem
A taxi charges £3 plus £1.50 per mile. How far can you travel for at most £18?
- 3 + 1.5m ≤ 18, so 1.5m ≤ 15 and m ≤ 10.
- Up to 10 miles.
Quadratic inequalities (Higher)
For a quadratic inequality, find the critical values by solving the quadratic equation, then decide which side of them works. A quick sketch of the parabola makes this clear.
Example 7: a quadratic inequality
Solve x² − x − 6 < 0.
- Solve x² − x − 6 = 0: (x − 3)(x + 2) = 0, so x = 3 or x = −2.
- The graph is a U shape, below the x-axis between the roots.
- −2 < x < 3.
Example 8: the outside region
Solve x² ≥ 16.
- x² = 16 gives x = ±4. The parabola y = x² − 16 is above the axis outside the roots.
- x ≤ −4 or x ≥ 4. A common mistake is writing only x ≥ 4.
Inequalities on graphs (Higher)
A linear inequality in two variables, such as y < 2x + 1, describes a region of the graph. Draw the boundary line y = 2x + 1, using a dashed line for < or > (not included) and a solid line for ≤ or ≥ (included). Then test a point not on the line, such as (0, 0): 0 < 1 is true, so the region containing the origin is the one you want. Questions often ask you to shade the region satisfying several inequalities at once, then list the integer coordinates inside it.
Example 9: integer points in a region
List the integer coordinates (x, y) that satisfy y ≥ 1, x < 3 and y < x + 2.
- Draw y = 1 (solid), x = 3 (dashed) and y = x + 2 (dashed).
- Check integer points: from y ≥ 1 and y < x + 2, we need x > y − 2, so for y = 1, x can be 0, 1 or 2; for y = 2, x can be 1 or 2; for y = 3, x = 2.
- (0, 1), (1, 1), (2, 1), (1, 2), (2, 2), (2, 3).
Example 10: fractions
Solve (2x + 1)/3 > x − 2.
- Multiply both sides by 3 (positive, so no flip): 2x + 1 > 3x − 6.
- Rearrange: 7 > x, so x < 7. Clearing fractions first keeps the algebra simple.
Why the answer is a range
An equation like 3x = 15 has one solution, but an inequality like 3x < 15 is true for infinitely many numbers. The solution x < 5 describes all of them at once. That is why exam questions ask you to show answers on number lines or list integers: they test whether you understand the whole set of solutions, not just the boundary value.
Common mistakes
- Forgetting to reverse the sign when multiplying or dividing by a negative.
- Using a filled circle for < or an open circle for ≤.
- Forgetting 0 or the negative integers when listing integer solutions.
- Giving only one region for a quadratic inequality like x² > 9.
- Using a solid boundary line for a strict inequality on a graph.
Practice questions
Try each one before opening the answer.
Q1. Solve 4x − 3 ≤ 13.
Show answer
Q2. Solve 5 − 3x < −10.
Show answer
Q3. List the integers satisfying 2 < 3n ≤ 12.
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Q4. Solve x² − 5x + 4 > 0.
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Frequently asked questions
Are inequalities on the Foundation paper?
Yes. Number lines, solving linear inequalities and listing integers are on both tiers. Quadratic inequalities and regions on graphs are Higher.
How do I check my answer?
Pick a value inside your solution and substitute it into the original inequality; it should be true. Then try a value just outside; it should be false.
Key takeaways
- Solve inequalities like equations, doing the same to both sides.
- Reverse the sign when multiplying or dividing by a negative.
- Open circle for < or >; filled circle for ≤ or ≥.
- For quadratics, find the roots and sketch the parabola.
- On graphs, dashed lines for strict inequalities; test a point to choose the region.