E2.6 Inequalities

Learning objectives

E2.6.1Represent and interpret inequalities, including on a number line.• Represent and interpret inequalities, including on a number line.E2.6.2Construct, solve and interpret linear inequalities.• Construct, solve and interpret linear inequalities.E2.6.3Represent and interpret linear inequalities in two variables• Represent and interpret linear inequalities in two variables graphically.E2.6.4List inequalities that define a given region• List inequalities that define a given region. 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 Examples include: • 3x < 2x + 4 • –3 ⩽ 3x – 2 < 7. The following conventions should be used: • broken lines should be used to represent strict inequalities (<, >) • solid lines should be used to represent inclusive inequalities (⩽, ⩾) • shading should be used to represent unwanted regions (unless otherwise directed in the question). e.g. 0 12 0 12 xx yy x < 1 y /uni2A7E E2.6.5Linear programming problems are not included.• Linear programming problems are not included.

An inequality describes a set of possible values rather than one value. Its symbol controls both the endpoint and the direction shown on a number line.

Inequality	Endpoint	Values shown
$x<a$	open circle at $a$	to the left
$x\le a$	closed circle at $a$	to the left
$x>a$	open circle at $a$	to the right
$x\ge a$	closed circle at $a$	to the right

$-2<x\le4$

The compound inequality means values greater than $-2$ and at most $4$ . Draw an open circle at $-2$ , a closed circle at $4$ , and one continuous segment between them.

An open circle excludes its endpoint; a closed circle includes it. The arrow or shaded segment shows the allowed values, not merely the direction in which the symbol points.

A linear inequality is solved with balance-preserving operations like a linear equation, except that multiplying or dividing both sides by a negative number reverses the inequality sign.

Define the unknown, translate phrases precisely—‘more than’ gives $>$ , ‘at least’ gives $\ge$ , ‘fewer than’ gives $<$ and ‘at most’ gives $\le$ —then check the units and context.

$4-3x\ge\frac{6-x}{5}\;\Longrightarrow\;20-15x\ge6-x\;\Longrightarrow\;-14x\ge-14\;\Longrightarrow\;x\le1$

The final sign reverses because both sides are divided by $-14$ . Adding or subtracting a negative number does not by itself reverse the sign.

If $1<x\le5$ and integer values are requested, list $2,3,4,5$ . Respect both endpoints and the stated number set.

Test one value inside the solution and one outside in the original inequality. This catches a reversed sign or an incorrectly included endpoint.

Do not replace $<$ by $\le$ when solving. An answer may need interpretation—for example, a whole-number count can require the least or greatest admissible integer.

A linear inequality in $x$ and $y$ describes one side of a boundary line. Several inequalities overlap to form a feasible region containing every point that satisfies all of them.

Inequality type	Boundary line
strict: $<$ or $>$	broken line; boundary excluded
inclusive: $\le$ or $\ge$	solid line; boundary included

Replace the inequality by an equality and draw its boundary; choose a test point not on the line; substitute it to decide which side satisfies the inequality; following the Cambridge convention, shade the unwanted side unless the question directs otherwise; repeat for every inequality and label the unshaded overlap $R$ .

$x\ge2,\qquad y\ge x,\qquad 2x+y\le8$

The boundaries $x=2$ , $y=x$ and $2x+y=8$ are all solid. A point in $R$ must lie right of $x=2$ , on or above $y=x$ , and on or below $2x+y=8$ .

Shading the unwanted region means the solution is the part left unshaded. Do not infer the correct side from the line’s gradient; use a test point.

To recover inequalities from a drawn region, identify each boundary equation, read whether the line is included, then determine which side contains the region.

Write the equation of each boundary; use $<$ or $>$ for a broken line and $\le$ or $\ge$ for a solid line; select a point clearly inside the region; substitute it to choose the correct sign; verify that every listed inequality contains the whole region.

Boundary	Equation form
vertical line through $a$	$x=a$
horizontal line through $b$	$y=b$
sloping line	find $y=mx+c$ or an equivalent form such as $ax+by=c$

If $R$ lies below a broken line $x+y=4$ , above a solid line $y=1.5$ , and below a solid line $y=2x+1$ , then $x+y<4$ , $y\ge1.5$ , and $y\le2x+1$ .

Line style decides strict versus inclusive; location decides the direction. A correct boundary equation with the wrong inequality sign does not define the same region.

For Cambridge IGCSE Mathematics E2.6, you must represent, solve and interpret inequalities and identify regions. Linear programming problems are explicitly not included.

Included	Not included
draw boundary lines with correct solid or broken style	formulate a business optimisation model
shade unwanted regions and identify the overlap	optimise an objective function systematically
read or list inequalities defining a region	use vertex testing as a general linear-programming procedure

A question may still ask you to read a largest or smallest value directly from a supplied region. Use the graph as directed, but do not add an unrequested linear-programming method or extend the syllabus into optimisation theory.

This card controls the assessable boundary; it does not create an additional calculation method or a new card type.

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E2.6 Inequalities

Learning objectives

Represent and interpret inequalities on a number line

Construct, solve and interpret linear inequalities

Represent inequalities in two variables graphically

List the inequalities that define a region

Stay within the inequalities scope: no linear programming

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