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Class 11 · Chapter 5 · Algebra unit (25 of 80 marks)

Linear Inequalities Class 11: notes and important questions

Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 17 questions
  • 5 multiple choice, 1 assertion–reason, 3 very short answer, 4 short answer, 2 long answer, 2 case study
  • About 6 hours to master

Algebra unit: 25 of 80 theory marks (Complex Numbers and Quadratic Equations, Linear Inequalities, Permutations and Combinations, Binomial Theorem, Sequences and Series).

Revision notes

Linear Inequalities: revision notes

1. Inequalities

Statements with \(\lt, \gt, \le, \ge\) are inequalities. \(ax + b \lt 0\) (and the other three forms) with \(a \neq 0\) is a linear inequality in one variable. The solution set is the set of all values of the variable that make it true; it depends on the set the variable comes from (\(\mathbb N\), \(\mathbb Z\) or \(\mathbb R\)).

2. Rules for solving

  • The same number may be added to or subtracted from both sides.
  • Both sides may be multiplied or divided by the same positive number.
  • Multiplying or dividing both sides by a negative number reverses the sign: \(-2x \ge 6 \Rightarrow x \le -3\).
  • Clear fractions by multiplying by the (positive) LCM of the denominators.

3. The number line

A solid dot marks an end-point that is included (\(\le, \ge\)), an open circle one that is not (\(\lt, \gt\)). In interval notation: \(x \lt 4\) is \((-\infty, 4)\); \(-1 \le x \lt 3\) is \([-1, 3)\).

4. Systems and double inequalities

Solve each inequality separately and take the values common to all (the intersection). A double inequality \(a \lt px + q \le b\) can be solved all at once by doing the same operation to the three parts.

5. Word problems

Translate "at least" as \(\ge\), "at most" / "not more than" as \(\le\), "more than" as \(\gt\). Check whether the answer must be a whole number (people, items) and state the final set of values.

Worked example 1

Solve \(5 - 2x \le 11\), \(x \in \mathbb R\).

\(-2x \le 6 \Rightarrow x \ge -3\): \([-3, \infty)\).

Worked example 2

Solve \(-5 \le \dfrac{2x - 1}{3} \lt 3\).

\(-15 \le 2x - 1 \lt 9 \Rightarrow -14 \le 2x \lt 10 \Rightarrow -7 \le x \lt 5\).

Worked example 3

Solve \(2x + 3 \gt 7\) for \(x \in \mathbb N\).

\(x \gt 2\): \(\{3, 4, 5, \ldots\}\).

Common errors

  • Not reversing the sign when dividing by a negative number.
  • Wrong end-points in the answer: \(\lt\) gives an open circle and a round bracket.
  • For a system, giving the union instead of the intersection.
  • Giving a decimal number of people: round in the direction the inequality allows.

Exam tips

  • Write the final answer as an interval (or a set for \(\mathbb N, \mathbb Z\)) and draw it on a number line when asked.
  • In word problems, write the inequality in words first, then in symbols.

Topics in this chapter: Rules for solving · Inequalities · Systems and double inequalities · Word problems.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can solve one-step and two-step linear inequalities and write the solution as an interval.

Read first: 1. Inequalities; 2. Rules for solving; 3. The number line 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%
Practise step 1
Step 2

Exam standard

You can clear fractions, solve double inequalities and systems, and show them on a number line.

Read first: 2. Fractions; 4. Systems and double inequalities; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%
Practise step 2
Step 3

Full marks on long answers

You can model budgets, marks and speeds with inequalities and give whole-number answers where needed.

Read first: 5. Word problems 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%
Practise step 3
Step 4

95+ stretch (HOTS)

You can solve systems of three inequalities and count integer solutions.

Read first: 4. Systems; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%
Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 4 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceRules for solving

The solution of \(3x - 5 \lt 7\), \(x \in \mathbb R\), is

  1. (a)\(x \lt 4\)
  2. (b)\(x \gt 4\)
  3. (c)\(x \lt \dfrac23\)
  4. (d)\(x \le 4\)
Q2·1 mark·Multiple choiceRules for solving

If \(-2x \ge 6\), then

  1. (a)\(x \ge -3\)
  2. (b)\(x \ge 3\)
  3. (c)\(x \le -3\)
  4. (d)\(x \le 3\)
Q3·1 mark·Multiple choiceInequalities

The solution set of \(2x + 1 \gt 5\) when \(x\) is a natural number is

  1. (a)\(\{2, 3, 4, \ldots\}\)
  2. (b)\((2, \infty)\)
  3. (c)\(\{1, 2\}\)
  4. (d)\(\{3, 4, 5, \ldots\}\)

Stretch yourself: 7 competition-style number theory problems (Senior, ages 16 to 18), original, with hints and full solutions. More extension & competition maths →

Where marks are lost in Linear Inequalities

  • Sign not reversed after dividing by a negative. Fix: circle the negative coefficient and flip the sign in the same line.
  • End-points shown wrongly on the number line. Fix: solid dot for ≤ / ≥, open circle for < / >.
  • Union given for a system. Fix: a system needs values satisfying every inequality: the overlap.
  • Fractions cleared with a negative multiplier or only on one side. Fix: multiply every term by the positive LCM.
  • Non-integer answers for counts of people or objects. Fix: list the whole numbers in the solution set.

Test Linear Inequalities against the clock

Want it against the clock? Take a timed 30-mark chapter test on Linear Inequalities, new questions each time (CBSE Essentials or the free trial).