Sec 3 · G3 (Express)

Inequalities

Practise inequalities with step-by-step help. Free account.

1. Solving inequalities

Step	Sign	Example
$+$ or $-$ a number	stays	$x + 3 > 5 \;\to\; x > 2$
$\times$ or $\div$ a positive	stays	$2x \leq 8 \;\to\; x \leq 4$
$\times$ or $\div$ a negative	reverses	$-2x \leq 8 \;\to\; x \geq -4$

Solve like an equation

\displaystyle ax + b > c

e.g.

\displaystyle 3x - 4 > 11

\displaystyle 3x > 15

\displaystyle x > 5

Negative x term: the sign reverses

\displaystyle -ax < c \;\Rightarrow\; x > -\dfrac{c}{a}

e.g.

\displaystyle 7 - 2x \leq 15

\displaystyle -2x \leq 8

\displaystyle x \geq -4

Fractions: multiply by the LCM

\displaystyle \dfrac{2x - 1}{3} > \dfrac{x + 2}{2}

e.g.

\displaystyle \dfrac{2x - 1}{3} > \dfrac{x + 2}{2}

\displaystyle 2(2x - 1) > 3(x + 2)

\displaystyle 4x - 2 > 3x + 6

\displaystyle x > 8

📖 Worked example

Solve $10 - 2x < -12$ .

Step 1 of 2

Collect the x terms on one side and the numbers on the other

-2x < -22

📖 Worked example

Solve $\dfrac{2x - 3}{2} \leq \dfrac{x + 1}{3}$ .

Step 1 of 4

Multiply both sides by the LCM of the denominators

3(2x - 3) \leq 2(x + 1)

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2. Number lines

Sign	Circle	End value
$<$ or $>$	open ○	not included
$\leq$ or $\geq$	filled ●	included

\displaystyle x \geq 2

\displaystyle x < -1

📖 Worked example

Write down the inequality shown on the number line.

Step 1 of 2

Read the circles

A filled circle means the value is included (

\leq

\geq

); an open circle means it is not (

<

>

Practise this →

3. Simultaneous inequalities

a < px + q ≤ b: do the same to all three parts

\displaystyle a < px + q \leq b

e.g.

\displaystyle -1 < 2x + 3 \leq 9

\displaystyle -4 < 2x \leq 6

\displaystyle -2 < x \leq 3

Two inequalities: solve each, then overlap

\displaystyle A \text{ and } B

e.g.

\displaystyle 2x - 3 > 1 \text{ and } 5 - x \geq 1

\displaystyle x > 2 \text{ and } x \leq 4

\displaystyle 2 < x \leq 4

Integers in a range

\displaystyle -2 < x \leq 3

$x = -1,\ 0,\ 1,\ 2,\ 3$
leave out $-2$ , include $3$

No overlap

\displaystyle x > 5 \text{ and } x < 2

e.g.

\displaystyle x > 5 \text{ and } x < 2

\displaystyle \text{no solution}

x in more than one part: split into two

\displaystyle x - 3 < 2x + 1 \leq 9

e.g.

\displaystyle x - 3 < 2x + 1 \text{ and } 2x + 1 \leq 9

\displaystyle x > -4 \text{ and } x \leq 4

\displaystyle -4 < x \leq 4

\displaystyle -2 < x \leq 3

\displaystyle 2 < x \leq 4

📖 Worked example

Solve $-35 \leq -4x - 5 \leq -17$ .

Step 1 of 3

Do the same to all three parts, so the x term is alone in the middle

-30 \leq -4x \leq -12

📖 Worked example

Solve $3x - 5 < -2$ and $2x + 3 > -5$ , or write "no solution" if there is none.

Step 1 of 4

Solve the first inequality

3x - 5 < -2

gives

x < 1

📖 Worked example

Solve the inequality $2x - 5 \leq \dfrac{2x - 1}{3} < x + 1$ .

Step 1 of 4

Split the chain into two inequalities: the middle part goes in both

2x - 5 \leq \dfrac{2x - 1}{3}

and

\dfrac{2x - 1}{3} < x + 1

Practise this →

4. Integer solutions

Inequality	Integers
$x < 6$	largest: $5$
$x \leq 6$	largest: $6$
$x > -4$	smallest: $-3$
$x \leq \dfrac{11}{4}$	largest: $2$

📖 Worked example

Find the smallest integer value of $x$ that satisfies $-7 - 2x < 1$ .

Step 1 of 3

Collect the x terms on one side and the numbers on the other

-2x < 8

📖 Worked example

Write down all the integer values of $x$ that satisfy $-5x + 2 < 12$ and $-3x + 5 > -1$ .

Step 1 of 5

Solve the first inequality

-5x + 2 < 12

gives

x > -2

$-3 \leq y \leq 2$	Integers	$y^2$
smallest $y^2$	$y = 0$	$0$
largest $y^2$	$y = -3$	$9$

📖 Worked example

$x$ and $y$ are integers such that $-6 \leq x < -3$ and $-2 < y \leq 2$ . Find the largest possible value of $y^2 - x^2$ .

Step 1 of 4

List the integers each letter can be

x

-6

-5

-4

y

-1

0

1

2

Practise this →

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Sec 3 · G3 (Express)

Inequalities

Practise inequalities with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Solving inequalities

Step	Sign	Example
$+$ or $-$ a number	stays	$x + 3 > 5 \;\to\; x > 2$
$\times$ or $\div$ a positive	stays	$2x \leq 8 \;\to\; x \leq 4$
$\times$ or $\div$ a negative	reverses	$-2x \leq 8 \;\to\; x \geq -4$

Solve like an equation

\displaystyle ax + b > c

e.g.

\displaystyle 3x - 4 > 11

\displaystyle 3x > 15

\displaystyle x > 5

Negative x term: the sign reverses

\displaystyle -ax < c \;\Rightarrow\; x > -\dfrac{c}{a}

e.g.

\displaystyle 7 - 2x \leq 15

\displaystyle -2x \leq 8

\displaystyle x \geq -4

Fractions: multiply by the LCM

\displaystyle \dfrac{2x - 1}{3} > \dfrac{x + 2}{2}

e.g.

\displaystyle \dfrac{2x - 1}{3} > \dfrac{x + 2}{2}

\displaystyle 2(2x - 1) > 3(x + 2)

\displaystyle 4x - 2 > 3x + 6

\displaystyle x > 8

📖 Worked example

Solve $10 - 2x < -12$ .

Step 1 of 2

Collect the x terms on one side and the numbers on the other

-2x < -22

📖 Worked example

Solve $\dfrac{2x - 3}{2} \leq \dfrac{x + 1}{3}$ .

Step 1 of 4

Multiply both sides by the LCM of the denominators

3(2x - 3) \leq 2(x + 1)

Practise this →

2. Number lines

Sign	Circle	End value
$<$ or $>$	open ○	not included
$\leq$ or $\geq$	filled ●	included

\displaystyle x \geq 2

\displaystyle x < -1

📖 Worked example

Write down the inequality shown on the number line.

Step 1 of 2

Read the circles

A filled circle means the value is included (

\leq

\geq

); an open circle means it is not (

<

>

Practise this →

3. Simultaneous inequalities

a < px + q ≤ b: do the same to all three parts

\displaystyle a < px + q \leq b

e.g.

\displaystyle -1 < 2x + 3 \leq 9

\displaystyle -4 < 2x \leq 6

\displaystyle -2 < x \leq 3

Two inequalities: solve each, then overlap

\displaystyle A \text{ and } B

e.g.

\displaystyle 2x - 3 > 1 \text{ and } 5 - x \geq 1

\displaystyle x > 2 \text{ and } x \leq 4

\displaystyle 2 < x \leq 4

Integers in a range

\displaystyle -2 < x \leq 3

$x = -1,\ 0,\ 1,\ 2,\ 3$
leave out $-2$ , include $3$

No overlap

\displaystyle x > 5 \text{ and } x < 2

e.g.

\displaystyle x > 5 \text{ and } x < 2

\displaystyle \text{no solution}

x in more than one part: split into two

\displaystyle x - 3 < 2x + 1 \leq 9

e.g.

\displaystyle x - 3 < 2x + 1 \text{ and } 2x + 1 \leq 9

\displaystyle x > -4 \text{ and } x \leq 4

\displaystyle -4 < x \leq 4

\displaystyle -2 < x \leq 3

\displaystyle 2 < x \leq 4

📖 Worked example

Solve $-35 \leq -4x - 5 \leq -17$ .

Step 1 of 3

Do the same to all three parts, so the x term is alone in the middle

-30 \leq -4x \leq -12

📖 Worked example

Solve $3x - 5 < -2$ and $2x + 3 > -5$ , or write "no solution" if there is none.

Step 1 of 4

Solve the first inequality

3x - 5 < -2

gives

x < 1

📖 Worked example

Solve the inequality $2x - 5 \leq \dfrac{2x - 1}{3} < x + 1$ .

Step 1 of 4

Split the chain into two inequalities: the middle part goes in both

2x - 5 \leq \dfrac{2x - 1}{3}

and

\dfrac{2x - 1}{3} < x + 1

Practise this →

4. Integer solutions

Inequality	Integers
$x < 6$	largest: $5$
$x \leq 6$	largest: $6$
$x > -4$	smallest: $-3$
$x \leq \dfrac{11}{4}$	largest: $2$

📖 Worked example

Find the smallest integer value of $x$ that satisfies $-7 - 2x < 1$ .

Step 1 of 3

Collect the x terms on one side and the numbers on the other

-2x < 8

📖 Worked example

Write down all the integer values of $x$ that satisfy $-5x + 2 < 12$ and $-3x + 5 > -1$ .

Step 1 of 5

Solve the first inequality

-5x + 2 < 12

gives

x > -2

$-3 \leq y \leq 2$	Integers	$y^2$
smallest $y^2$	$y = 0$	$0$
largest $y^2$	$y = -3$	$9$

📖 Worked example

$x$ and $y$ are integers such that $-6 \leq x < -3$ and $-2 < y \leq 2$ . Find the largest possible value of $y^2 - x^2$ .

Step 1 of 4

List the integers each letter can be

x

-6

-5

-4

y

-1

0

1

2

Practise this →

Try a question: Inequalities

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise inequalities with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free