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Algebra · Subtopic 5 of 11

Inequalities and Regions

Inequalities are solved almost like equations, with one trap: multiplying or dividing by a negative flips the sign. On a graph, an inequality in $x$ and $y$ is a region on one side of a line: solid line if included, dashed if not.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Solve linear and double inequalities.
Show solutions on a number line and list integer solutions.
Draw the region represented by a set of inequalities.
Find points with integer coordinates in a region.

Pause and think

$3<5$ is true. Multiply both sides by $-1$ : is $-3<-5$ true?

Section 2 of 14

Core idea

Start hereA range, not a number

"You must be under 5 to travel free." That's not one age but all ages below 5. Inequalities describe ranges, and solving them is like solving equations, with one twist.

The big idea

Solve like equations, but flip the sign when multiplying or dividing by a negative. Regions: boundary lines (solid for ≤, dashed for <) plus a test point.

1. Linear inequalities

Number line for −2 ≤ x < 3

A filled circle means the end value is allowed (

\le

); an open circle means it is not (

<

). Integers in the range:

-2,-1,0,1,2

Why the sign flips

$2<3$
True.
$-2>-3$
Multiplying by $-1$ reflects the number line, reversing order.

Worked exampleThree inequalities

$3x-4<11$ ; $5-2x>11$ ; $-3<2x+1\le7$ (list integers).

Goal: Solutions.
Strategy: Balance; flip on negatives.

$3x<15\Rightarrow x<5$
Add 4, divide by 3.
$-2x>6\Rightarrow x<-3$
Divide by $-2$ : flip.
$-4<2x\le6\Rightarrow-2<x\le3$
Do the same to all three parts.

Answer

x<5

;

x<-3

; integers

-1,0,1,2,3

✓ Check

x=-4

5+8=13>11

✓.

2. Regions

The region y ≥ 1, x + y ≤ 6, y ≤ 2x

Shade the region that satisfies all three (here shown shaded as the wanted region; some exams shade the unwanted parts instead, so read the question).

Draw each boundary: solid for $\le,\ge$ ; dashed for $<,>$ .
Test a point (e.g. the origin) to choose the side.

Common misconception $x^2<9$ Not just

x<3

: also

x>-3

. Answer

-3<x<3

Pause and think

Does $(0,0)$ satisfy $y>2x+1$ ?

Summary

Flip when × or ÷ by a negative.
Regions: boundary, test point.

Inequality: Statement with $<,>,\le,\ge$ .

HistoryHarriot's symbols. The signs

<

and

>

first appeared in a book by Thomas Harriot published in

1631

, ten years after his death. Harriot was also an astronomer who drew the Moon through a telescope months before Galileo.

Fun factAirlines run on inequalities. Deciding how many seats to sell at each price is a huge system of inequalities (fuel, weight, crew, demand). The method of finding the best corner of a region, called linear programming, earned a Nobel Prize in Economics in

1975

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