Above the line: x² > 4
x² = 4 at x = −2 and x = 2
y = x² is above y = 4 outside them
x < −2 or x > 2
Test x = −3: 9 > 4, true
Quadratic inequalities Why is the answer to x² > 9 two inequalities, not just x > 3?
Higher · AQA · 45 minutes
Students find where a curve is above a line before anyone explains, then solve quadratic inequalities by sketching the curve through its roots and reading off where it is above or below the axis. Slides to present, a vote that shows who's stuck, practice and homework.
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For which x is the curve y = x² above the line y = k? Move k and write the answer for k = 1 and k = 4. What changes when k goes below 0?
They cross at (−2, 4) and (2, 4)
The idea in one minute
To solve a quadratic inequality, get 0 on one side and sketch the curve through its roots. Where the expression is more than 0, the curve is above the x-axis; where it is less than 0, below. For a curve with +x², below the axis is one piece between the roots, a < x < b, and above the axis is two pieces outside them, x < a or x > b: two inequalities, joined by or.
The lesson goes picture first. Students find where y = x² is above a line, and see the answer come in two pieces, so x > 2 alone can't be all of it. Then they factorise, as for a quadratic equation, sketch, and read off. Testing one number in each piece checks every answer. It is Higher; the stretch turns the curve upside down.
By the end, students can
- explain why x² > 9 has two pieces to its answer
- solve a quadratic inequality by sketching the curve through its roots
- write the answer as one inequality or two, with < or ≤ as the question has it
- check an answer by testing a number in each piece
- count the integers that satisfy a quadratic inequality
Key words
- quadratic inequality
- roots
- sketch
- above the axis
- below the axis
- or
Before this lesson
Students should solve a quadratic by factorising (x² − 2x − 8 = 0), sketch y = x², and solve a linear inequality such as 2x + 1 < 7.
Taught in Quadratic equations.
The lesson, step by step
30 minutes in total. Each step says what changes for this length.
3 min 1 · Hook · students think
Where is the curve above the line?
Students find where y = x² is above a line before anyone explains.
For this length: Ask about x = −3 at once; skip the silent minutes
Say“Here are y = x² and the line y = 4. For which x is the curve above the line? Find as many as you can, then describe all of them.”
Ask- Is x = −3 one of them? Is (−3)² more than 4?
- How many pieces is your answer in?
Watch forWriting x > 2 only: ask about x = −3, where x² is 9.
For the teacher
Give two minutes of silent work, then collect answers. Many will write x > 2, as if solving x² = 4.
Don't explain yet. The question is the point: x = −3 works too, so what does the whole answer look like?
Collect the answers as written. Keep x > 2 on its own on the board: the next step is about the piece it misses.
Shorter: Ask about x = −3 at once; skip the silent minutes (−2 min).
8 min 2 · Explain · you explain
Sketch, find the roots, read off
Above or below the axis decides the answer; the roots are its ends; one inequality or two.
For this length: Skip the ≥ slide; say that ≤ and ≥ include the roots
Say“Get 0 on one side. Sketch the curve through its roots. More than 0 is where the curve is above the axis; less than 0 is where it is below.”
Ask- x² − 2x − 8 < 0: where is the curve below the axis?
- Can a number be less than −2 and more than 2 at once?
Watch forSquare-rooting as for an equation, and losing the negative piece.
Below the axis: x² − 2x − 8 < 0
(x − 4)(x + 2) < 0
Roots: x = −2 and x = 4
Below the axis between them: −2 < x < 4
Test x = 0: −8 < 0, true
On or above the axis: x² + x − 6 ≥ 0
(x + 3)(x − 2) ≥ 0
On or above the axis outside the roots
x ≤ −3 or x ≥ 2
For the teacher
First slide: y = x² and the line y = 4 meet at x = −2 and x = 2. The curve is above the line on both outer sides: x < −2 or x > 2. Test x = −3: 9 > 4. So x > 2 alone misses half of the answer. The slide shades the x that work; its edges are dashed, as −2 and 2 are not in, just as for a region drawn with <.
Second slide: get 0 on one side, then factorise, as for a quadratic equation. x² − 2x − 8 < 0 is (x − 4)(x + 2) < 0. The curve is below the axis between its roots: −2 < x < 4, one inequality.
Third slide: ≥ 0 is on or above the axis: outside the roots, two pieces, x ≤ −3 or x ≥ 2. With ≤ and ≥ the roots are in, and the shading's edges are solid; with < and > they are not, and the edges are dashed.
Fourth slide: one piece is one inequality, a < x < b; two pieces are two, joined by or. −2 > x > 2 would say x is below −2 and above 2 at once: no number is.
Vote before each reading (Ask the class): one piece or two?
Shorter: Skip the ≥ slide; say that ≤ and ≥ include the roots (−4 min).
5 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
For this length: Do the first two pairs only
Say“Watch me do one. Then you do its twin, which is almost the same.”
Ask- Above or below the axis? So one piece or two?
Watch forDividing x² > 3x by x to get x > 3: x could be negative, and the piece x < 0 is lost.
Against a number
x² < 16
x² = 16 at x = −4 and x = 4
Below y = 16 between them: −4 < x < 4
Then theirs
x² ≥ 25
Factorise first
x² − 5x + 6 > 0
(x − 2)(x − 3) > 0
Above the axis outside the roots: x < 2 or x > 3
Then theirs
x² + 2x − 15 ≤ 0
Get 0 on one side
x² > 3x, so x² − 3x > 0
x(x − 3) > 0: roots 0 and 3
Above the axis outside them: x < 0 or x > 3
Then theirs
x² ≤ 2x + 8
For the teacher
Work the first example aloud, writing each line and sketching the curve. Then students do its twin while you watch who writes one inequality for two pieces.
Do the same for each pair. The third pair catches students who divide by x, or read off before getting 0 on one side.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
9 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: After the vote, questions 1–4 only
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Test a number in each piece of your answer: does it work?
Watch forChoosing without testing a number: ask for one number their answer includes, and check it.
Sketch each curve through its roots and show your working on paper. Test a number in each piece. The stretch is optional.
The question that shows who's stuck
- Solve x² > 49.
In your Classroom, as they answer (an example) - A9
- B11
- C3
- D4
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- x² < 36, so ___ < x < ___
- (x − 1)(x − 6) ≤ 0, so ___ ≤ x ≤ ___
- x² − 4x − 12 > 0, so x < ___ or x > ___
- x² ≥ 6x, so x ≤ ___ or x ≥ ___
- Which values of x satisfy x² < −4?
- Zara writes the answer to x² − 1 > 0 as −1 > x > 1. What is wrong?
- How many integers satisfy x² − 3x − 10 < 0?
- 2x² + 5x − 3 > 0, so x < ___ or x > ___
- Stretch: 6 + x − x² > 0, so ___ < x < ___
Support · For question 3: factorise x² − 4x − 12, sketch the curve through its roots, then ask where it is above the axis. How many pieces?
For the teacher
Everyone chooses before anyone explains. Then ask every student, whatever they chose: which of −10, 0 and 10 make x² > 49 true? Does your answer include exactly those? Name one number your answer includes.
Questions 1–2 build fluency; 3–4 change the look; 5–6 ask for reasons; 7–8 are exam-style. The stretch turns the curve upside down.
Students sketch and show their working on paper or the board; the quiz keeps their final answers.
If the votes are split, test −10, 0 and 10 together in x² > 49, and sketch y = x² with the line y = 49, before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: what changes when x² has a minus sign? (+10 min).
5 min 5 · Exit · students show
Solve one, say why
One inequality solved and one sentence show who reads the sketch rather than a rule.
Say“Solve it, sketch the curve, and say why your answer is one inequality or two.”
Watch forWriting two pieces as one joined inequality.
Show your working and your sketch; one sentence for the last part.
- Solve x² − x − 12 ≥ 0. Sketch the curve, and say in one sentence why your answer is one inequality or two.
For the teacher
Solve x² − x − 12 ≥ 0 and sketch the curve through its roots.
Then say in one sentence why the answer is one inequality or two.
The sentences show who reads the sketch: start the next lesson with the hook's curve and line if few do. Set the homework.
Longer: Count the integers that do not satisfy it, and say why (+5 min).
One to one: ask the student to sketch before writing anything, then to test one number in each piece of their answer and one number outside it.
Homework: sketch and read off
Sketch each curve through its roots before you write the answer. Test one number in each piece of your answer.
- Solve x² ≤ 64.
- Solve x² − 7x + 10 < 0.
- Solve x² + 3x − 18 ≥ 0.
- Solve x² > 5x. Why must you not divide both sides by x?
- List every integer that satisfies x² − 2x − 15 < 0.
- Sketch y = x² + 4. Explain why every x satisfies x² + 4 > 0, and no x satisfies x² + 4 < 0.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching quadratic inequalities: common questions
Why is x² > 9 not just x > 3?
Because negative numbers square to positives: x = −5 gives 25, more than 9. On a graph, y = x² is above the line y = 9 on both outer sides, left of −3 and right of 3. Square-rooting as for an equation keeps one side only; a sketch shows both.
How do students know whether the answer is one inequality or two?
From the sketch, never a rule to learn. With +x², the curve is below the axis in one piece between the roots (a < x < b) and above it in two pieces outside them (x < a or x > b). With −x² it is the other way round, which is why the sketch is worth the ten seconds.
Should students sketch the curve or use a sign table?
Either gives the answer; the sketch shows why, and is quicker to check. This lesson sketches every time and tests one number in each piece. A sign table is a good second method for students going on to A level.
Is quadratic inequalities on the Foundation or Higher tier?
Higher. Students need to solve quadratic equations by factorising and solve linear inequalities first. The stretch, with −x², is the hardest case GCSE asks for.
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Curriculum reference: AQA GCSE Mathematics 8300: A22 (quadratic inequalities in one variable, Higher), with A11 (roots from a graph) and A18 (solving quadratics by factorising).
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