Linear inequalities

Foundation and Higher

The idea

Solve an inequality like an equation, but if you multiply or divide by a negative number, flip the sign. $-2x < 6$ gives $x > -3$ .

Rearrange to get $x$ on its own.
Flip the inequality only when multiplying or dividing by a negative.
On a number line, a closed circle means included ( $\le$ or $\ge$ ), an open circle means not.

Watch out Integers satisfying $-2 < x \le 3$ are $-1, 0, 1, 2, 3$ : check the ends.

Worked exampleNon-calculator3 marks

Answer each part.

(a) Solve $3x - 5 < 16$

(b) $n$ is an integer and $-2 < n \le 3$ . Write down all the possible values of $n$ .

(a) Add 5: $3x < 21$ . Divide by 3: $x < 7$ .
(b) $-2$ is not included ( $<$ ), and $3$ is included ( $\le$ ). So $n$ can be $-1, 0, 1, 2, 3$ .

Answer (a) $x < 7$ ; (b) $-1, 0, 1, 2, 3$

Question 1Non-calculator3 marks

Solve $4(x - 1) \le 20$ .

(a) Solve the inequality.

(b) Write down the largest integer that satisfies the inequality.

Question 2Non-calculator4 marks

$-4<2x+1\le 13$ , where x is an integer.

(a) Write all possible values of x.

(b) Find the sum of all these values.

Question 3Non-calculator6 marks

Answer each part.

(a) Solve $-7 \le 3x + 2 < 11$

(b) $x$ is an integer. It satisfies both $4x - 1 > 2x + 6$ and $\frac{x}{2} < 5$ . How many possible values of $x$ are there?

83 official questions in 80 papers since 2017. The latest:

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