CA Foundation · Quantitative Aptitude

Linear Inequalities for CA Foundation: Chapter Guide

A linear inequality compares two expressions using <, >, ≤ or ≥, such as 2x + 3 ≤ 11. You solve it like an equation, but reverse the sign when you multiply or divide by a negative number. The answer is a range of values, shown on a number line or graph.

What this chapter covers

Linear Inequalities is a chapter in Business Mathematics, part of Paper 3, Quantitative Aptitude. It deals with statements like x + 5 > 9 instead of x + 5 = 9. The answer is usually a set of values, not one number. You learn to find that set, write it as an interval, and show it on a number line or a graph.

The chapter has two parts. First is one variable: solve, then write the solution set. Second is two variables: draw a boundary line, shade the correct half-plane, and find the region that satisfies several inequalities together. Word problems then turn a situation, such as a budget or a minimum requirement, into inequalities.

This chapter builds on the equations chapter, because the solving steps are almost the same. It also builds on the basics of coordinate graphs and straight lines, which you need to draw boundary lines and shade regions. If your equation skills are weak, fix them first. Most errors in this chapter come from algebra slips, not from the new ideas.

Paper 3 is an objective paper with 0.25 negative marking, so you need accurate and quick answers. This chapter rewards that. Most questions have one clean method, and wrong options can often be removed by substituting a single value. Once you know the sign rule and the shading test, you can answer in under a minute. Because the rules are few, it is a good chapter to master early and rely on in the exam. The skills also help you in other Business Mathematics topics that use ranges and graphs.

Linear Inequalities: topics in the order to study them

1Inequalities and Their TypesStart here to learn the symbols, strict versus slack inequalities, and what a solution set means before you solve anything.
2Rules for Solving Linear InequalitiesThese rules, especially the sign reversal for negatives, are used in every later topic, so learn them before practice.
3Solving Inequalities in One VariableThis applies the rules to real problems and teaches you to write answers as intervals and on a number line.
4Linear Inequalities in Two VariablesYou extend to two variables by drawing a boundary line and shading a half-plane, which needs the one-variable skills first.
5System of Linear InequalitiesA system is several two-variable inequalities at once, so you can only do it after you can graph a single one.
6Word Problems and ApplicationsStudy this last because you need all the earlier tools to turn a statement into inequalities and solve them.

How to prepare Linear Inequalities

Aim for a clean method you can repeat fast, then test it with timed MCQs. Do not memorise answers. Practise the steps until they are automatic.

Write the four symbols and what each means on one page. Note that < and > are strict, while ≤ and ≥ include the boundary value.
Learn the rules: you can add or subtract the same number on both sides, and multiply or divide by a positive number, without changing the sign. Multiply or divide by a negative number and the sign reverses.
Solve 15 to 20 one-variable problems. After each, pick one value from your answer and substitute it to check. Do this until checking takes ten seconds.
For two variables, practise the same routine: draw the boundary as a solid line for ≤ or ≥ and a dashed line for < or >, then test the point (0, 0) to choose the side. If the line passes through (0, 0), test another point.
For systems, find the overlap of all shaded regions. Practise reading the corner points from the intersection of boundary lines.
For word problems, define the variables first, write each condition as an inequality, and add any non-negativity conditions such as x ≥ 0.
Finish with timed MCQ sets. In the exam, use substitution on the options and skip any question where the setup takes too long, because wrong answers cost 0.25 marks.

Common mistakes in Linear Inequalities

Forgetting to reverse the sign when dividing by a negative number.
Fix: Circle every negative coefficient you divide by. Then check with a test value, such as in −2x < 6, x = 0 works, so the answer must be x > −3.
Using the wrong bracket or including an end value that is not allowed.
Fix: Link the symbol to the bracket: < or > gives ( ), and ≤ or ≥ gives [ ]. Infinity always takes a round bracket.
Shading the wrong side of the line in two-variable problems.
Fix: Always substitute a test point that is not on the line. If the line goes through the origin, use a point such as (1, 0) or (0, 1).
Using a solid line for strict inequalities, or a dashed one for slack inequalities.
Fix: Remember: if the symbol has an equals bar (≤ or ≥), the line is solid. If not, it is dashed.
Setting up word problems with the wrong inequality direction.
Fix: Translate them in a fixed way: at least means ≥, at most or not more than means ≤, more than means >, less than means <.
Ignoring conditions like x ≥ 0 in applied problems.
Fix: Ask whether the quantity can be negative. If not, add the non-negativity condition before finding the region or the options that fit.

Last-day revision: Linear Inequalities

Symbols: < less than, > greater than, ≤ at most, ≥ at least.
Adding or subtracting the same number on both sides keeps the sign.
Multiplying or dividing both sides by a positive number keeps the sign.
Multiplying or dividing both sides by a negative number reverses the sign.
Strict inequalities (< or >) exclude the end value; round bracket ( ) in interval form.
Slack inequalities (≤ or ≥) include the end value; square bracket [ ] in interval form.
Always use round brackets for infinity, never square.
Boundary line of ax + by < c or > c is dashed; for ≤ or ≥ it is solid.
Test (0, 0) to find the shaded side, unless the line passes through it.
For a system, the solution is the common region of all inequalities.
In word problems, add x ≥ 0 and y ≥ 0 when quantities cannot be negative.
Check your answer by substituting one value from the solution set.

Linear Inequalities practice questions

Linear Inequalities: frequently asked questions

Is Linear Inequalities an important chapter for CA Foundation?

It is part of Business Mathematics in Paper 3, and its methods are simple and repeatable. Once you know the rules, questions are quick to solve. It is a good chapter to get right so you do not lose marks to negative marking.

What is the most important rule in linear inequalities?

When you multiply or divide both sides by a negative number, the inequality sign reverses. Many wrong answers come from missing this. Always verify with a test value from your solution set.

How do I graph a linear inequality in two variables?

First draw the boundary line by treating the inequality as an equation. Use a solid line for ≤ or ≥ and a dashed line for < or >. Then test a point not on the line, such as (0, 0), and shade the side where the test is true.

How can I save time on MCQs from this chapter?

Substitute values from the options into the inequality instead of solving fully. A single well-chosen value can remove two or more options. If a word problem looks long, solve other questions first and return to it, as a wrong answer costs 0.25 marks.