B7PureCore

Linear and Quadratic Inequalities

Q: What does interval notation like (-, -2) (3, ) mean?

It is the same as “x 3”. Round brackets exclude the endpoint (strict inequality); square brackets include it ( or ); means “or” (the union of the two pieces).

An inequality describes a range of values rather than a single answer. Linear ones behave almost like equations — with one crucial twist — while quadratic ones can only be solved safely by thinking about the shape of the graph.

28 min Video by TLMaths Algebra and Functions

The big picture

Most real questions are inequalities, not equations: not “when does the profit equal zero?” but “for which prices is the profit positive?”. The honest difficulty of this topic is that a quadratic inequality cannot be solved by blindly rearranging — the answer is a $region$ , and which region depends on whether the parabola opens up or down. Students who memorise “solve and flip” get quadratic inequalities wrong constantly; students who sketch the parabola get them right every time. So this lesson is really about replacing a fragile rule with a reliable picture. It also completes the discriminant story: those “find the values of $k$ ” problems finish as quadratic inequalities.

A-Level Maths B5-12: Examples of Solving Quadratic InequalitiesWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Solve linear inequalities, flipping the sign when multiplying or dividing by a negative
Solve quadratic inequalities using critical values and a sketch
Decide correctly between an “outside” region and a “between” region
Express solutions with inequalities or interval notation

Linear inequalities and the one rule that changes

You solve a linear inequality exactly like an equation — add, subtract, multiply, divide to isolate $x$ — with a single exception: $multiplying or dividing by a negative number reverses the inequality sign$ . This is because negating flips order: $3 < 5$ , but $- 3 > - 5$ .

So treat the inequality like an equation, and stay alert for that one moment where a negative multiplier or divisor appears.

a < b \Rightarrow - a > - b

Multiplying by a negative flips the direction.

1Subtract 4:

- 2 x > 6

2Divide by

- 2

— and flip the sign:

x < - 3

Answer

x < - 3

Tip — If you dislike flipping signs, keep the $x$ term positive instead: from $4 - 2 x > 10$ , add $2 x$ to both sides to get $4 > 10 + 2 x$ , then solve normally. Same answer, no flip to forget.

Quadratic inequalities: find the critical values

A quadratic inequality such as $x^{2} - x - 6 > 0$ cannot be solved by rearranging to “ $x >$ something”. Instead, follow three steps. First, solve the matching $equation$ ( $x^{2} - x - 6 = 0$ ) to find the $critical values$ — where the curve crosses the $x$ -axis. Second, $sketch$ the parabola. Third, read off the region that satisfies the inequality.

The critical values split the number line into pieces; the sketch tells you which pieces lie above the axis ( $y > 0$ ) and which lie below ( $y < 0$ ).

1Solve the equation:

x^{2} - x - 6 = (x - 3) (x + 2) = 0

, so critical values

x = 3

and

x = - 2

2Sketch: the parabola opens upward (positive

x^{2}

) and crosses at

- 2

and

3

3We want

y > 0

— where the curve is

above

the axis — which is

outside

the roots.

Answer

x < - 2

x > 3

Why this works

For an upward parabola, $y > 0$ (above the axis) happens $outside$ the roots, and $y < 0$ (below the axis) happens $between$ them. The sketch encodes this instantly — which is why you should always draw it rather than trust a remembered rule.

Outside vs between — and writing the answer

The most common error is giving one region when the answer needs two, or vice versa. A “ $> 0$ ” (upward parabola) answer is two separate pieces, written as $x < a$ or $x > b$ . A “ $< 0$ ” answer is the single stretch between the roots, written as one chain $a < x < b$ .

You may write solutions either as inequalities ( $x < - 2$ or $x > 3$ ) or in $interval notation$ : $(- \infty, - 2) \cup (3, \infty)$ , where a round bracket excludes the endpoint. Use square brackets and $\leq / \geq$ when the endpoints are included.

1Same critical values

x = - 2

and

x = 3

2Now we want

y \leq 0

— where the curve is on or below the axis — which is

between

the roots.

3Endpoints included because of “

\leq

”.

Answer

- 2 \leq x \leq 3

(interval:

[- 2, 3]

)

Tip — A “between” answer $a < x < b$ is a single statement. Never write an “outside” answer as $3 < x < - 2$ — that chain is impossible and signals the regions are reversed.

Think like an examiner

A sketch of the parabola (roots and direction of opening) is expected working for a quadratic inequality, and often carries a mark — draw it every time.
Match the region to the sign: “ $> 0$ ” with an upward parabola is $outside$ the roots (two pieces); “ $< 0$ ” is $between$ (one piece).
Never write an impossible chain like $3 < x < - 2$ ; an “outside” region must be given as “ $x < - 2$ or $x > 3$ ”.
Watch strictness: use $<, >$ for strict inequalities and $\leq, \geq$ when endpoints are included.

Common misconceptions

Forgetting to reverse the sign when dividing by a negative, e.g.

- 2 x > 6 \Rightarrow x > - 3

Dividing by a negative flips the inequality:

x < - 3

Solving a quadratic inequality by rearranging to a single “

x >

…”.

Find the critical values, sketch the parabola, and read off the region — quadratics give regions, not a single bound.

Writing an “outside” solution as one chain, e.g.

3 < x < - 2

Two separate regions must be written “

x < - 2

x > 3

”. A chain is only for a genuine “between”.

Choosing “between” when the inequality is “

> 0

” (or vice versa).

Let the sketch decide: above the axis is outside the roots (for an upward parabola); below is between.

Inequality essentials

\div (-) \Rightarrow flip the sign

The linear twist.

(x - a) (x - b) > 0 \Rightarrow x < a or x > b

Upward parabola: outside the roots.

(x - a) (x - b) < 0 \Rightarrow a < x < b

Upward parabola: between the roots.

Stretch yourself

The equation $x^{2} + k x + 4 = 0$ has two distinct real roots. Find the range of values of $k$ .

Hint — Two distinct real roots means discriminant $> 0$ . This gives a quadratic inequality in $k$ — solve it with a sketch.

Questions students ask

When do I flip the inequality sign?

Only when you multiply or divide both sides by a negative number. Adding, subtracting, or multiplying/dividing by a positive never changes the direction. To avoid the flip entirely, rearrange so the

x

term stays positive.

Why can’t I solve a quadratic inequality like a normal equation?

Because the solution is a range, and which range depends on the shape of the parabola. Rearranging to a single “

x >

…” loses that shape. Find the critical values, sketch, and read off the correct region.

How do I know whether the answer is “between” or “outside” the roots?

Sketch the parabola. For an upward-opening curve, it is above the axis (

> 0

) outside the roots and below the axis (

< 0

) between them. A downward parabola reverses this — the sketch always tells you.

What does interval notation like

(- \infty, - 2) \cup (3, \infty)

mean?

It is the same as “

x < - 2

x > 3

”. Round brackets exclude the endpoint (strict inequality); square brackets include it (

\leq

\geq

);

\cup

means “or” (the union of the two pieces).

Key takeaways

Solve linear inequalities like equations, but flip the sign when multiplying/dividing by a negative.
For quadratics: find critical values, sketch the parabola, read off the region.
Upward parabola: “ $> 0$ ” is outside the roots; “ $< 0$ ” is between them.
Write “outside” as two regions ( $x < a$ or $x > b$ ) — never as an impossible chain.
Discriminant conditions on an unknown finish as quadratic inequalities.

How this fits the course

Build on

Graphs and Transformations (later)soon

Leads to

Polynomials and the Factor Theorem

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