Inequalities and Regions
An equation has a few exact answers; an inequality usually has a whole range of them. "x is bigger than 3" is true for 3.1, 4, 100 and infinitely many other numbers. This page shows how to write, draw and solve inequalities, and (Higher) how to solve quadratic inequalities and shade regions on a graph.
GCSE subject content: solve linear inequalities in one variable; represent the solution set on a number line; Higher: solve linear inequalities in two variables and quadratic inequalities in one variable; represent the solution set using set notation and on a graphBefore you start. Solving an inequality uses the same "do the same to both sides" steps as solving a linear equation (see Linear equations). For the regions section you'll need to draw straight lines like y = 2x + 1 (see Linear graphs).
1 · Symbols and number lines
There are four inequality symbols. The open end of the symbol faces the bigger number, like a mouth eating it.
| Symbol | Means | Example | On a number line |
|---|---|---|---|
| < | is less than | x < 3 | open circle at 3, arrow left |
| ≤ | is less than or equal to | x ≤ 3 | filled circle at 3, arrow left |
| > | is greater than | x > 3 | open circle at 3, arrow right |
| ≥ | is greater than or equal to | x ≥ 3 | filled circle at 3, arrow right |
An open circle means the end number is not included (it's a strict inequality); a filled circle means it is included. Try the builder, then test yourself.
Which inequality does this number line show?
2 · Solving linear inequalities
To solve an inequality means to get x on its own, like solving an equation. You may add or subtract the same number on both sides, and multiply or divide both sides by a positive number, and the inequality stays true. There is one exception, and it's the whole point of this section:
multiply or divide by a negative → flip the sign
Why? 2 < 5 is true. Multiply both by −1: −2 and −5. But −2 is bigger than −5. Multiplying by a negative reflects every number in 0, so the order reverses. Slide k and watch the two plum numbers swap order.
Ways to avoid the flip. In 7 − 2x > 1 you can add 2x to both sides first: 7 > 1 + 2x, so 6 > 2x, so 3 > x, which is the same as x < 3. Either way, check with a number: x = 0 gives 7 > 1 ✓ and 0 < 3 ✓.
Never write the answer as an equation (x = 3) or forget which way it points. "3 > x" and "x < 3" say the same thing; read them carefully.
3 · Double inequalities and integer solutions
A double inequality like −2 < x ≤ 4 traps x between two values: bigger than −2 and at most 4. Solve it by doing the same thing to all three parts. An integer is a whole number (positive, negative or zero), and questions often ask you to list the integers that satisfy an inequality. Watch the ends: is each one included?
4 · Quadratic inequalities Higher
Set notation is a compact way to write a solution: {x : −2 < x < 3} reads "the set of all x such that x is between −2 and 3". For two separate pieces, write {x : x < −2} ∪ {x : x > 3} (∪ means "or", the union of the two sets).
To solve a quadratic inequality like x² − x − 6 < 0, never try to "solve it like an equation". Instead:
- Solve the equation x² − x − 6 = 0 to find the critical values (here x = −2 and x = 3).
- Sketch the graph y = x² − x − 6: a U shape crossing the x-axis at those values.
- "< 0" means "where is the curve below the x-axis?" That's between the roots. "> 0" means above: the two outside pieces.
Trap: x² < 9. It's tempting to write x < ±3, but that is meaningless. Sketch y = x² − 9: it is below the axis between −3 and 3, so −3 < x < 3. And x² > 9 is x < −3 or x > 3 (two separate inequalities, not one).
5 · Inequalities on a graph Higher
An inequality with two letters, like y > 2x − 1, is true for a whole region of the coordinate grid. The line y = 2x − 1 is the boundary. Every point above it has a bigger y, so satisfies y > 2x − 1; every point below doesn't.
- Draw the boundary dashed for < or > (points on the line are not included) and solid for ≤ or ≥ (they are).
- To decide which side, use a test point not on the line, usually (0, 0): substitute it. True → shade that side; false → the other side.
Drag the plum test point around and watch the substitution.
6 · Regions from several inequalities Higher
When a question gives several inequalities, the region R that satisfies all of them is where all the shaded parts overlap. Draw each boundary line (solid or dashed), decide which side is wanted, and the region in the middle is usually a triangle or other polygon. Questions often ask you to list the integer points in R: points whose coordinates are both whole numbers. Points on a dashed boundary don't count; points on a solid one do.
Click every integer point inside R, then press Check.
Worked examples
Solve 3x + 5 < 20 and show the solution on a number line
Solve 7 − 2x ≥ 1
List the integers n with −3 < 2n + 1 ≤ 9
Find the smallest integer x with 5x − 2 > 2x + 7
Solve x² − 2x − 15 ≤ 0, giving your answer in set notation Higher
Solve x² > 16 Higher
Exam-style: list the integer points (x, y) satisfying y ≥ 1, x + y < 6 and y ≤ 2x Higher
Quick check
What you've learned
- < and > are strict (open circle, end not included); ≤ and ≥ include the end (filled circle).
- Solve inequalities like equations, but flip the sign when you multiply or divide by a negative, because that reverses the order of numbers.
- For a double inequality, do the same to all three parts. When listing integers, check each end: included or not?
- Higher Quadratic inequalities: find the critical values, sketch the curve, then read off where it is below (< 0, between the roots) or above (> 0, two outside pieces). Write answers in set notation, e.g. {x : −2 < x < 3}.
- Higher Regions: boundary lines are dashed for <, > and solid for ≤, ≥; use a test point to choose the side; the required region satisfies every inequality at once.