Sec 3 · G2 (N(A))

Inequalities

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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

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. 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

Practise this →

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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

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. 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

Practise this →