C2.6 Inequalities

Learning objectives

C2.6.1Represent and interpret inequalities, including on a number line• Represent and interpret inequalities, including on a number line. When representing and interpreting inequalities on a number line: • open circles should be used to represent strict inequalities (<, >) • closed circles should be used to represent inclusive inequalities (⩽, ⩾) e.g. – 3 ⩽ x < 1 –/uni202F3– /uni202F2– /uni202F101 x

An inequality describes a set of possible values rather than one value. Its symbol tells both the boundary and whether the boundary value itself belongs to the set.

Inequality	Read as	Endpoint on a number line	Direction from the endpoint
$x<a$	$x$ is less than $a$	open circle	left
$x>a$	$x$ is greater than $a$	open circle	right
$x\le a$	$x$ is at most $a$	closed circle	left
$x\ge a$	$x$ is at least $a$	closed circle	right

An open circle means the endpoint is excluded, matching $<$ or $>$ . A closed circle means the endpoint is included, matching $\le$ or $\ge$ . The line or ray shows all the other permitted values.

A bounded interval combines two conditions. The syllabus example $-3\le x<1$ means values from $-3$ up to but not including 1: use a closed endpoint at $-3$ , an open endpoint at 1, and join the points between them.

When reading a number line, inspect each endpoint before writing the compound inequality. A closed point at $-5$ and open point at 1 with the interval between them gives $-5\le x<1$ ; keep the smaller boundary on the left.

If only integer solutions are requested, list the integers inside the interval. For $-3\le x<3$ , they are $-3,-2,-1,0,1,2$ : include $-3$ because of $\le$ , but exclude 3 because of $<$ .

Do not decide endpoint inclusion from the direction of the line: inclusion depends only on whether the circle is closed. This Core objective represents and interprets given inequalities; solving inequalities algebraically is outside this card.

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