MathsCBSE Class 11 · Linear Inequalities

Solving Linear Inequalities in One Variable: Rules, Fractions and Double Inequalities

Q: Why does the inequality sign flip?

Multiplying by a negative number reflects the number line about zero and reverses the order of any two numbers.

Q: How is the solution different in N, Z and R?

In R it is an interval; in Z or N it is a list of integers or natural numbers within the interval.

Q: Do I need to graph inequalities in two variables?

The chapter page notes that graphical solutions of inequalities in two variables are assessed only formatively in 2026-27.

Q: What if the coefficient of x contains a parameter?

Work out the case in which the sign of the coefficient is fixed by the form of the given answer, as in the worked example with a.

Updated 17 October 2026 · Fact-checked

Treat a linear inequality like an equation, with one change: multiplying or dividing both sides by a negative number reverses the sign. Clear fractions by multiplying by the LCM of the denominators (positive), collect x on one side, and write the answer as an interval. A double inequality a < f(x) < b is solved by applying each operation to all three parts at once. The solution set depends on the number system (N, Z or R) asked for.

Dividing both sides by −3 turns > into <, so the answer is every number left of −3, with an open circle because −3 itself is excluded.

Illustration drawn for QuestionBuddy.

Understand Solving Linear Inequalities: Rules, Fractions and Double Inequalities

A linear inequality in one variable compares two linear expressions with $<, \leq, >, \geq$ . Its solution set is the set of all numbers in the stated domain that make it true.

Rules. 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 by a negative number reverses the sign: $- 3 x > 9 \Rightarrow x < - 3$ . Never multiply by an expression whose sign you do not know.

Solving routine.
1. Remove brackets and fractions (multiply by the positive LCM).
2. Collect the $x$ terms on one side and numbers on the other.
3. Divide by the coefficient of $x$ (reverse the sign if it is negative).
4. State the answer in interval notation and mark it on the number line: an open circle for $<, >$ and a filled dot for $\leq, \geq$ .

Dependence on the domain. $x < 5$ gives $(- \infty, 5)$ in $R$ , but ${\dots, 3, 4}$ in $Z$ and ${1, 2, 3, 4}$ in $N$ . Always read the domain in the question, and list integer or natural solutions explicitly when asked.

Double inequalities. To solve $- 5 \leq \frac{5 - 3 x}{2} \leq 8$ , multiply all three parts by $2$ , subtract $5$ from all three, then divide by $- 3$ , reversing both signs: you end with $- \frac{11}{3} \leq x \leq 5$ . Write the final inequality with the smaller number on the left. The number of integers in $[- \frac{11}{3}, 5]$ is nine, from $- 3$ to $5$ .

Parameters. If the question says "the solution set is $x \geq 5$ , find $k$ ", solve the inequality in terms of $k$ and equate the boundary values. Watch the sign of the coefficient of $x$ : if it depends on a parameter, discuss the cases or use the given direction of the answer.

Compound statements. "Both" means intersection (the common part of the two solution sets); "either" means union.

Why the rule reverses. On the number line, $2 < 5$ but $- 2 > - 5$ : multiplying by $- 1$ reflects points about $0$ and flips their order.

Solving Linear Inequalities: Rules, Fractions and Double Inequalities: formulas to remember

Add or subtract: $a < b g i v es a + c < b + c$; Direction unchanged.
Multiply by positive: $a < ban d c > 0 g i v es a c < b c$; Direction unchanged.
Multiply by negative: $a < ban d c < 0 g i v es a c > b c$; Direction reversed.
Double inequality: $a < f (x) < b : o p er a t eo na l l t h r ee p a r t s$; Reverse both signs when dividing by a negative.
Intervals: $(a, b) o p e n; [a, b] c l ose d; [a, b) an d (a, b] ha l f - o p e n$; Open circle for strict, filled dot for non-strict.

How to solve Solving Linear Inequalities: Rules, Fractions and Double Inequalities questions

Use these steps for any linear inequality, including those with fractions and double inequalities.

1Read the domain ( $N$ , $Z$ or $R$ ) in the question.
2Clear brackets; multiply by the positive LCM of the denominators to remove fractions.
3Move variable terms to one side and constants to the other.
4Divide by the coefficient of $x$ , reversing the sign if it is negative.
5Write the solution set in interval notation and show it on the number line.
6Test one number inside the interval and one outside in the original inequality.

Solving Linear Inequalities: Rules, Fractions and Double Inequalities in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A budget inequality

Suppose a cab company charges Rs 120 as the base fare and Rs 14 per km, and you have Rs 400 to spend.

What distance $d$ (in km) can you travel?

Try it first, then show the solutionSolution

Cost $= 120 + 14 d \leq 400$ .
$14 d \leq 280$ , so $d \leq 20$ .
Distance cannot be negative, so $0 \leq d \leq 20$ km.

Answer: At most 20 km.

Memory hook: Budget means "less than or equal to".

A double inequality from a temperature range

Suppose a vaccine fridge must stay between $2^{\circ}$ C and $8^{\circ}$ C, and a display shows Fahrenheit using $F = \frac{9}{5} C + 32$ .

In what range should the Fahrenheit reading lie?

Try it first, then show the solutionSolution

$2 \leq C \leq 8$ . Multiply all parts by $\frac{9}{5}$ (positive): $3.6 \leq \frac{9}{5} C \leq 14.4$ .
Add $32$ : $35.6 \leq F \leq 46.4$ .

Answer: $35. 6^{\circ} F \leq F \leq 46. 4^{\circ} F$

Memory hook: Do the same operation on all three parts of a double inequality.

Tricks to remember Solving Linear Inequalities: Rules, Fractions and Double Inequalities

Short memory aids. Say each one aloud once, then use it on a question the same day.

Flip on negative
Multiply or divide by a negative: flip the sign. Add or subtract: never flip.
Use it when: Every inequality.
Smaller on the left
Write the final double inequality as small number, then x, then large number.
Use it when: Presentation of the answer.
LCM of denominators
Multiply every part by the same positive LCM so no fraction remains.
Use it when: Fractions.
Domain first
Say whether the answer is an interval or a list of integers.
Use it when: N, Z, R questions.

Solving Linear Inequalities: Rules, Fractions and Double Inequalities: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Brackets on both sides · 2 marks · Very short answerBoard level

Solve $3 (2 - x) \geq 2 (1 - x)$ for real $x$ and write the answer as an interval.

Try it first, then show the solutionSolution

$6 - 3 x \geq 2 - 2 x$ . (½ mark)
$4 \geq x$ , i.e. $x \leq 4$ . (1 mark)
Solution set $(- \infty, 4]$ . (½ mark)

Answer: $(- \infty, 4]$

Example 2: Fractions · 3 marks · Short answerBoard level

Solve $\frac{3 x - 4}{2} \geq \frac{x + 1}{4} - 1$ for $x \in R$ , and list the solutions that are natural numbers up to 4.

Try it first, then show the solutionSolution

Multiply by 4: $2 (3 x - 4) \geq (x + 1) - 4$ . (1 mark)
$6 x - 8 \geq x - 3$ , so $5 x \geq 5$ , $x \geq 1$ . (1 mark)
Solution set $[1, \infty)$ ; natural solutions up to 4: $1, 2, 3, 4$ . (1 mark)

Answer: $[1, \infty)$ ; $1, 2, 3, 4$

Example 3: A double inequality · 3 marks · Short answerHarder board level

Solve $- 5 \leq \frac{5 - 3 x}{2} \leq 8$ and find how many integers satisfy it.

Try it first, then show the solutionSolution

Multiply by 2: $- 10 \leq 5 - 3 x \leq 16$ . (½ mark)
Subtract 5: $- 15 \leq - 3 x \leq 11$ . (½ mark)
Divide by $- 3$ , reversing both signs: $5 \geq x \geq - \frac{11}{3}$ , i.e. $- \frac{11}{3} \leq x \leq 5$ . (1 mark)
Integers from $- 3$ to $5$ : nine of them. (1 mark)

Answer: $[- \frac{11}{3}, 5]$ ; 9 integers

Example 4: Finding a parameter · 3 marks · Short answerHarder board level

Find the value of $a$ for which the solution set of $a x + 3 > 2 x - 5$ is $x > - 4$ .

Try it first, then show the solutionSolution

Rewrite: $(a - 2) x > - 8$ . (1 mark)
For the solution to be of the form $x > \dots$ , we need $a - 2 > 0$ ; then $x > \frac{- 8}{a - 2}$ . (1 mark)
Equate: $\frac{- 8}{a - 2} = - 4$ , so $a - 2 = 2$ , $a = 4$ (which satisfies $a > 2$ ). (1 mark)

Answer: $a = 4$

Example 5: Parameter in a fractional inequality · 4 marks · Short answerHOTS / competency

If the solution set of $\frac{2 x - 1}{3} + 1 \geq \frac{x + k}{2}$ is exactly $x \geq 5$ , find $k$ .

Try it first, then show the solutionSolution

Multiply by 6: $2 (2 x - 1) + 6 \geq 3 (x + k)$ . (1 mark)
$4 x - 2 + 6 \geq 3 x + 3 k$ , so $4 x + 4 \geq 3 x + 3 k$ . (1 mark)
$x \geq 3 k - 4$ . (1 mark)
Compare with $x \geq 5$ : $3 k - 4 = 5$ , $k = 3$ . Check: with $k = 3$ , at $x = 5$ both sides equal $4$ . (1 mark)

Answer: $k = 3$

Where students lose marks in Solving Linear Inequalities: Rules, Fractions and Double Inequalities

The slips examiners see most often, and what to write instead.

Mistake: Not reversing the sign
Division by a negative number is treated like division by a positive one.
Wrong working
$- 3 x > 9 \Rightarrow x > - 3$ .
Right working
$x < - 3$ .
Do this: Circle every negative coefficient and write "reverse" next to it.
Usually costs: The whole answer.
Mistake: Reversing only one sign in a double inequality
Both inequality signs flip together.
Wrong working
$- 15 \leq - 3 x \leq 11 \Rightarrow 5 \leq x \leq - \frac{11}{3}$ .
Right working
$5 \geq x \geq - \frac{11}{3}$ , i.e. $- \frac{11}{3} \leq x \leq 5$ .
Do this: Rewrite with the smaller number on the left.
Usually costs: 1 to 2 marks.
Mistake: Forgetting the domain
An interval is given when the question asks for integers or natural numbers.
Wrong working
$x < 5$ for $x \in N$ answered as $(- \infty, 5)$ .
Right working
${1, 2, 3, 4}$ .
Do this: Underline the domain in the question.
Usually costs: The final mark.
Mistake: Multiplying only some terms by the LCM
The constant or one fraction is left unscaled.
Wrong working
$\frac{x + 1}{4} - 1$ becomes $(x + 1) - 1$ after multiplying by 4.
Right working
$(x + 1) - 4$ .
Do this: Put every term in brackets before multiplying.
Usually costs: The answer.

Exam tips

Write the domain ( $R$ , $Z$ or $N$ ) at the start of your answer.
Show the sign reversal as a separate line; examiners award a method mark for it.
Give the final answer in both interval notation and as an inequality when asked to represent it on the number line.
For "number of integer solutions", list the end integers to avoid off-by-one errors.
Test one value from the solution set in the original inequality as a check.

Practice questions on Solving Linear Inequalities: Rules, Fractions and Double Inequalities

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The solution set of $- 2 x + 7 \geq 1$ in $R$ is:
(a) $x \geq 3$
(b) $x \geq - 3$
(c) $x \leq 3$
(d) $x \leq - 3$
Hint
Divide by $- 2$ and reverse the sign.
Answer
(c). $- 2 x \geq - 6$ , so $x \leq 3$ .
Q2 · 1 mark · MCQHarder board level
The number of integers $x$ satisfying $- 3 \leq 2 x + 1 < 9$ is:
(a) 5
(b) 6
(c) 7
(d) 8
Hint
Subtract 1, then divide by 2.
Answer
(b). $- 4 \leq 2 x < 8$ , so $- 2 \leq x < 4$ : integers $- 2, - 1, 0, 1, 2, 3$ .
Q3 · 3 marks · Short answerHarder board level
Solve $\frac{2 ( x + 3 )}{5} - \frac{x - 1}{2} < 4$ and show the solution on the number line.
Hint
Multiply by 10.
Answer
$4 (x + 3) - 5 (x - 1) < 40$ gives $17 - x < 40$ , so $x > - 23$ . Interval $(- 23, \infty)$ with an open circle at $- 23$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If $a > b$ and $c > d$ , then $a - c > b - d$ .
Reason (R): Inequalities in the same direction can be added, but not subtracted.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Test with numbers such as $a = 5, b = 4, c = 10, d = 1$ .
Answer
(d). Take $a = 5, b = 4, c = 10, d = 1$ : $a - c = - 5$ and $b - d = 3$ , so $a - c < b - d$ ; A is false. R is true.
Q5 · 4 marks · Short answerHOTS / competency
Solve $x + 5 < 4 x - 7 \leq 2 x + 9$ for real $x$ . List the integer solutions and find their sum and mean.
Hint
Split into two inequalities and take the common part.
Answer
$x + 5 < 4 x - 7 \Rightarrow x > 4$ ; $4 x - 7 \leq 2 x + 9 \Rightarrow x \leq 8$ . Solution $(4, 8]$ . Integers $5, 6, 7, 8$ ; sum $26$ ; mean $6.5$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What happens to the sign when you multiply by a negative number?Show answer
It reverses.
2.What does a filled dot mean on the number line?Show answer
The end point is included (≤ or ≥).
3.How do you clear fractions?Show answer
Multiply every term by the positive LCM of the denominators.
4.How do you solve a < f(x) < b?Show answer
Apply each operation to all three parts.
5.What is the interval for x ≥ 1?Show answer
[1, ∞).

Solving Linear Inequalities: Rules, Fractions and Double Inequalities: frequently asked questions

Why does the inequality sign flip?

Multiplying by a negative number reflects the number line about zero and reverses the order of any two numbers.

How is the solution different in N, Z and R?

In R it is an interval; in Z or N it is a list of integers or natural numbers within the interval.

Do I need to graph inequalities in two variables?

The chapter page notes that graphical solutions of inequalities in two variables are assessed only formatively in 2026-27.

What if the coefficient of x contains a parameter?

Work out the case in which the sign of the coefficient is fixed by the form of the given answer, as in the worked example with a.