Chapter 5 · Introduction

Chapter 5 — Linear Inequalities

An equation asks for the one value that fits exactly. Real life usually asks something else — at least how many, at most how much — and the answer is a whole stretch of numbers. Linear inequalities are the tool for those questions.

Most of the algebra you have done so far has been about equations: find the value of $x$ that makes $3x-5$ equal to $7$. But look at the questions people actually ask. A lift carries at most $544$ kg; the operator weighs $64$ kg and each carton weighs $30$ kg — how many cartons can go in one trip? You have scored $72$, $65$ and $81$ in three unit tests and want an average of at least $75$ after the fourth — what must you score? A vaccine must be kept between $2\,^\circ\text{C}$ and $8\,^\circ\text{C}$; a recharge plan must cost no more than your pocket money; a JEE cut-off asks for a percentile above some mark. None of these has a single answer. "Any number of cartons up to $16$" and "any mark from $82$ to $100$" are the honest answers, and they are statements about ranges of numbers.

The symbols $\lt$, $\gt$, $\le$ and $\ge$ turn such questions into mathematics, and the resulting statements — inequalities — can be solved almost exactly like equations. Almost: there is one rule that catches out students every year. Multiply both sides of $2\lt5$ by $-1$ and you get $-2$ and $-5$, but now $-2$ is the bigger one, so the sign must turn round: $-2\gt-5$. In this chapter you will learn what an inequality is and when a number solves it, the rules that keep inequalities true (and the one that flips them), how to solve linear inequalities in one variable step by step, how to write the solution set as an interval and draw it on the number line, and how to set up and solve word problems about marks, budgets, temperatures and mixtures. In Class 12 the same ideas grow into linear programming, where companies use inequalities to plan production at the lowest cost.

Compare: $\lt$, $\gt$, $\le$, $\ge$→Rules: add freely; multiplying by a negative flips the sign→Solve $ax+b\lt c$→Answer as an interval on the number line→Systems and word problems
🎯 By the end of this chapter you can…
  • read and write strict ($\lt$, $\gt$) and slack ($\le$, $\ge$) inequalities, and recognise a linear inequality in one variable;
  • decide whether a number is a solution, and explain why the solution set depends on whether $x$ is a natural number, an integer or a real number;
  • apply the rules for inequalities correctly — adding or subtracting any number, multiplying or dividing by a positive number, and reversing the sign when multiplying or dividing by a negative number;
  • solve linear inequalities in one variable, including ones with brackets and fractions and ones with $x$ on both sides, and recognise when there is no solution or every real number is a solution;
  • solve a system of linear inequalities and a double inequality such as $-5\le 3-2x\lt 9$;
  • show every solution set on the number line and in interval notation, and translate word problems ("at least", "at most", "not more than") into inequalities and solve them.
Teaser — What Must I Score?

Your first three unit-test marks (out of $100$) are $72$, $65$ and $81$. You want the average of all four tests to be at least $75$. Move the fourth mark $x$ and watch the average line rise towards the target line. Below, the number line collects every mark that works: they form a whole stretch, $82\le x\le100$ — not one single answer.

60
total of four—
average—
average ≥ 75?—
all marks that work—

Sections in this chapter

5.1

Inequalities and the Rules for Handling Them

Strict and slack inequalities, linear inequalities, solutions and solution sets, and the rules that keep an inequality true — including the sign flip.

$a\lt b,\ c\lt0\ \Rightarrow\ ac\gt bc$
Open section →5.2

Solving Linear Inequalities in One Variable

Solving step by step, the number line and interval notation, systems and double inequalities, and word problems.

$3x-5\lt7\iff x\in(-\infty,4)$
Open section →
✅ Before you start
You need set notation and the number systems $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$ (Section 1.1), and above all intervals — $(a,b)$, $[a,b)$, $(-\infty,b]$ and so on, with hollow and filled end points on the number line — from Section 1.2: every answer in this chapter is written as one. The intersection of two sets (Section 1.3) is used for systems of inequalities. From earlier classes you need to solve linear equations in one variable confidently (including ones with brackets and fractions, as in Class 8 and the Class 10 chapter "Pair of Linear Equations in Two Variables"), and to be comfortable with negative numbers on the number line. The optional JEE box in Section 5.2 also uses the modulus function $|x|$ from Section 2.3.
← 4.3 The Argand Plane5.1 Inequalities and the Rules for Handling Them →