GCSE · Maths · AQA · Spec 8300 · Higher

Solving linear (one or two variables) and quadratic inequalities (Higher)

Is x² − 2x − 8 negative? At x = 1 it's −9; at x = 5 it's +7. So ask instead: which values of x make it negative?

Which values of x make x² − 2x − 8 negative?

-4-1.513.56-10-341118xy = x² − 2x − 8(1, -9)

x: 1. y = x² − 2x − 8: -9

You start at the bottom of the U, where y = −9. Drag the point left and right and keep an eye on the sign of y. Where exactly does it stop being negative?

Red line: y = 0: Every point on the red line has height 0. Above it, x² − 2x − 8 is positive; below it, negative. So the question “which x make it negative?” becomes “where is the curve below the red line?” — and you answer it by reading x-values along the bottom.

Exam line: x² − 2x − 8 < 0 is one stretch: −2 < x < 4. x² − 2x − 8 > 0 is two pieces running away from each other: x < −2 or x > 4. Same curve, opposite sign, very different answer.
Watch out: At x = −2 and x = 4 the expression is exactly 0. Zero isn't less than 0 and isn't greater than 0, so for < and > those two values are left out. For ≤ or ≥ they are included.

Start simple: linear inequalities

Solve it, then draw the answer

Solve −3 < 2x + 1 ≤ 7. A linear inequality works like an equation: do the same thing to every part. Then set the two ends of your answer and decide, for each end, whether that value is included.

Exam line: Read {x : −2 < x ≤ 3} as “the set of all x such that x is more than −2 and less than or equal to 3”. The colon means “such that”.

The one trap

Find the line that goes wrong

Solve 11 − 3x ≥ 2.

A student's answer — which line goes wrong?

Maths · Algebra

Back to the curve: the written method

Step through it one line at a time. Each line says what was done and why.

GoalSolve x² + x ≥ 12 and write the answer in set notation.
1
x² + x ≥ 12
start

The right-hand side isn't 0 yet. The curve trick only works when you're asking whether something is above or below 0, so that comes first.

2
3
4
5
6
7

Step 1 of 7

The right-hand side isn't 0 yet. The curve trick only works when you're asking whether something is above or below 0, so that comes first.

Watch out: Never squash two pieces into one line like 3 ≤ x ≤ −4. That says x is at least 3 and at most −4 at the same time — no number does that. Two pieces need “or”, or ∪ in set notation.

Now with x and y

Which points are in the region?

A region R is defined by y > x and x + y ≤ 6. On a graph, y = x would be drawn dashed and x + y = 6 solid. For each point, substitute its x and y into each inequality and tick every one it satisfies.

  • A y > x
  • B x + y ≤ 6
  1. (1, 4)
  2. (2, 2)
  3. (1, 5)
  4. (0, 7)
  5. (4, 1)
  6. (3, 3)
  7. (5, 4)
  8. (−1, 2)

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

An equation asks “what is x?”. An inequality asks a better question: “which values of x work?” — and you can see the answer on a line or a graph.

What you need to know

  • The solution of an inequality is a set of values. x > 2 means every number bigger than 2, not just 3.
  • Solve a linear inequality like an equation. Adding, subtracting, and multiplying or dividing by a positive number keep the sign the same. Multiplying or dividing both sides by a negative number reverses it.
  • On a number line, an open circle means the end value is not included (< or >); a filled circle means it is included (≤ or ≥). An answer like x > 2 is an open circle at 2 with an arrow pointing right.
  • Set notation: {x : −2 < x ≤ 3} means “the set of x such that −2 < x ≤ 3”. Two separate pieces are joined with ∪, for example {x : x ≤ −4} ∪ {x : x ≥ 3}.
  • For a quadratic inequality: rearrange so one side is 0, factorise, find the critical values (where the expression is 0), sketch the curve, then read off where it is below 0 (< 0) or above 0 (> 0).
  • When a U-shaped curve crosses the x-axis at two different roots, the expression is negative between them and positive outside them. So “< 0” gives one interval and “> 0” gives two pieces.
  • An inequality in x and y is shown as a region on a graph. The boundary line is dashed for < or > and solid for ≤ or ≥. The region for several inequalities is where all of them are true at once.

The big picture

Solving an inequality gives a set of values, not one number. Solve a linear inequality like an equation, but reverse the sign whenever you multiply or divide by a negative. Show the answer on a number line — open circle for < or >, filled for ≤ or ≥ — or in set notation. For a quadratic, get 0 on one side, factorise, sketch, and read off where the curve is below or above the axis. For a U that crosses at two roots, that means between the roots, or outside them. With x and y, the answer is a region bounded by dashed or solid lines.

Key points

1Ask “which values work?”, not “what is x?”. The answer is a set.
2Dividing or multiplying by a negative number turns the sign round. Adding and subtracting never do.
3Open circle = not included (< or >). Filled circle = included (≤ or ≥). Dashed line = not included. Solid line = included.
4Quadratics: get 0 on one side, factorise, find the critical values, sketch. Below the axis is “< 0”, above it is “> 0”.
5Check any answer by testing a value from inside it in the original inequality.

Worked example

Problem

Find all the points with integer coordinates that satisfy x ≥ 0, y > 1 and x + y ≤ 4.

⚠ Watch out

Solving x² > 16 like an equation and writing just x > 4. But x = −5 works too: (−5)² = 25, and 25 > 16. y = x² − 16 is above the axis on both sides, so the answer is x < −4 or x > 4.

🧠

Memory hook

Zero on one side, find where it crosses, then ask: is the curve under the line or over it? For a U that crosses twice: under = in between, over = out on the ends.

✓

Check yourself

Solve 7 − 2x > 1 and describe the answer on a number line. (Answer: −2x > −6, so x < 3 — the sign reverses. Open circle at 3, arrow pointing left.)

Flashcards

(12)
What kind of thing is the solution of an inequality?
A set of values — a whole range of numbers that all make the inequality true, not one single number.
When does the inequality sign reverse?
When you multiply or divide both sides by a negative number. Adding or subtracting never reverses it.
On a number line, what do an open circle and a filled circle mean?
Open circle: that end value is not included (< or >). Filled circle: it is included (≤ or ≥).
How do you show x ≥ −1 on a number line?
A filled circle at −1 with an arrow pointing to the right.
How do you read {x : 0 ≤ x < 5}?
“The set of all x such that x is at least 0 and less than 5.” The colon means “such that”.
What does the symbol ∪ mean in an answer such as {x : x < 1} ∪ {x : x > 6}?
It joins two pieces: x can be in either one. Use it when the answer is two separate stretches.
What is the first step in solving a quadratic inequality like x² + 3x > 10?
Rearrange so one side is 0 (x² + 3x − 10 > 0), then factorise to find the critical values.
What are the critical values of a quadratic inequality?
The values of x that make the expression exactly 0 — the roots, where the curve crosses the x-axis. They split the line into stretches.
A U-shaped curve crosses the x-axis at two different roots. Where is the expression negative and where is it positive?
Negative between the two roots (the curve dips below the axis). Positive outside them, on both sides.
On a graph, when is a boundary line dashed and when is it solid?
Dashed for < or > — points on the line are not included. Solid for ≤ or ≥ — points on the line are included.
How do you decide which side of a boundary line is the region you want?
Test a point that isn't on the line, such as (0, 0). If it makes the inequality true, that's the right side; if not, it's the other side.
A region is defined by several inequalities. Which points are in it?
Only the points that satisfy every inequality at the same time — the overlap.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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