Linear Inequalities Class 11: Notes, Number Lines and Solved Questions

What Linear Inequalities is about

Weightage: This chapter is part of Unit II, Algebra, which the syllabus gives 25 of the 80 theory marks in the school annual exam (CBSE adds that there is no fixed chapter-wise weightage).Source: CBSE Mathematics (041) syllabus 2026-27, Class XI course structure, Unit II Algebra

Key concepts

Inequalities and their kinds

NCERT section 5.2

Two real numbers or algebraic expressions joined by $<$, $>$, $\le$ or $\ge$ form an inequality.

• Strict inequalities use $<$ or $>$; slack inequalities use $\le$ or $\ge$.
• Numerical: only numbers, e.g. $3<5$. Literal: contains variables, e.g. $2x+3\ge 7$.
• Double inequality: two signs at once, e.g. $3<x<5$.
• Linear in one variable: $ax+b<0$ (or >, ≤, ≥), $a\ne 0$. Linear in two variables: $ax+by\le c$. Quadratic: contains $x^2$, e.g. $x^2-4>0$.

A solution is a value of the variable that makes the inequality true; the set of all of them is the solution set. For example, if a pen costs ₹12 and you have ₹50, the number of pens n you can buy satisfies $12n\le 50$, n a whole number.

Worked example

Classify each: (a) $2x-5>3$ (b) $7\le 9$ (c) $x^2+x\ge 2$ (d) $3x+2y<6$ (e) $-1\le x<4$. [2 marks]

Try it first, then show the solutionSolution

1. (a) literal, strict, linear in one variable; (b) numerical, slack
2. (c) quadratic, slack; (d) linear in two variables, strict; (e) double inequality

Answer: As classified above.

Rules for solving

NCERT section 5.3

Two rules are enough for every linear inequality:

Rule 1: Adding or subtracting the same number on both sides keeps the inequality true and keeps the sign. So terms may be moved across the sign with a change of sign, just as in equations.

Rule 2: Multiplying or dividing both sides by the same positive number keeps the sign. Multiplying or dividing by a negative number reverses it: $<$ becomes $>$, $\le$ becomes $\ge$, and so on.

Why Rule 2 flips: $2<5$, but $-2>-5$. Multiplying by −1 reflects the numbers in 0 on the number line, which reverses their order.

Never multiply both sides by an expression whose sign you do not know (such as x).

Worked example

Solve $7x-5\le 4x+10$ and $3-2x>11$ for real x. [2 marks]

Try it first, then show the solutionSolution

1. $7x-4x\le 10+5\Rightarrow 3x\le 15\Rightarrow x\le 5$, i.e. $(-\infty ,5]$
2. $-2x>8$; dividing by −2 reverses the sign: $x<-4$, i.e. $(-\infty ,-4)$

Answer: $x\le 5$; $x<-4$

Solutions in N, Z and R

NCERT section 5.3

The same inequality can have different solution sets depending on the allowed values of x.

1. First solve over R as usual, e.g. get $x<\frac{25}{6}$.
2. Then pick the allowed values:

• x in N: list the natural numbers that fit, here {1, 2, 3, 4}.
• x in Z: all integers that fit, here {…, −2, −1, 0, 1, 2, 3, 4}.
• x in R: the interval $(-\infty ,\frac{25}{6})$.

The solution set may be empty: $-15x>40$ gives $x<-\frac{8}{3}$, which no natural number satisfies.

Worked example

Solve $18x<75$ when (i) x is a natural number (ii) x is an integer. [2 marks]

Try it first, then show the solutionSolution

1. $x<\frac{75}{18}=\frac{25}{6}\approx 4.17$
2. (i) {1, 2, 3, 4} (ii) {…, −3, −2, −1, 0, 1, 2, 3, 4}

Answer: (i) {1, 2, 3, 4} (ii) all integers up to 4.

Inequalities with fractions and brackets

NCERT section 5.3

1. Expand brackets carefully (watch the sign in front of a bracket).
2. Clear fractions by multiplying both sides by the LCM of the denominators, which is positive, so the sign stays.
3. Collect x-terms on one side and numbers on the other.
4. Divide by the coefficient of x, reversing the sign if it is negative.
5. Write the answer as an inequality, an interval, and (if asked) on the number line.

Check: pick one value inside your answer and one outside, and test them in the original inequality.

Worked example

Solve $\frac{x-2}{3}-\frac{x+1}{4}\ge 1$ for real x. [2 marks]

Try it first, then show the solutionSolution

1. Multiply by 12: $4(x-2)-3(x+1)\ge 12\Rightarrow 4x-8-3x-3\ge 12$
2. $x-11\ge 12\Rightarrow x\ge 23$, i.e. $[23,\infty )$

Answer: $x\ge 23$

Showing solutions on the number line

NCERT section 5.3

For a real solution set:

• Open circle at a: a is not included ($<$ or $>$).
• Filled dot at a: a is included ($\le$ or $\ge$).
• Shade (draw a thick line) to the left for $x<a$ or $x\le a$, and to the right for $x>a$ or $x\ge a$.
• For $a<x\le b$, shade between a and b with an open circle at a and a filled dot at b.

For integer or natural solutions, mark only the separate points (dots), not a continuous line.

Interval link: $x<a$ is $(-\infty ,a)$; $x\ge a$ is $[a,\infty )$; $a<x\le b$ is $(a,b]$.

Worked example

Solve $5x+2<8x+11$ and show the solution on the number line. [2 marks]

Try it first, then show the solutionSolution

1. $2-11<8x-5x\Rightarrow -9<3x\Rightarrow x>-3$
2. Open circle at −3, thick line to the right: the interval $(-3,\infty )$

Answer: $x>-3$, i.e. $(-3,\infty )$

Double inequalities and systems

NCERT section 5.3 (Miscellaneous Examples)

Double inequality $p<ax+b\le q$: do the same operation on all three parts at once. Dividing by a negative number reverses both signs; then rewrite it with the smaller number on the left.

System of inequalities: solve each one separately, then take the values common to both (the intersection). A number line makes this easy: shade each solution and keep the overlap. If there is no overlap, the system has no solution.

Modulus as a double inequality (useful extra): for $a>0$, $|x-c|<a$ means $-a<x-c<a$.

Worked example

Solve $-7\le \frac{3-2x}{5}<1$. [3 marks]

Try it first, then show the solutionSolution

1. Multiply all parts by 5: $-35\le 3-2x<5$
2. Subtract 3: $-38\le -2x<2$
3. Divide by −2 and reverse both signs: $19\ge x>-1$, i.e. $-1<x\le 19$

Answer: $(-1,19]$

Word problems

NCERT section 5.3 (Examples 7, 8 and Miscellaneous)

Translate phrases carefully:

• 'at least' → $\ge$; 'at most', 'not more than' → $\le$
• 'more than', 'exceeds' → $>$; 'less than', 'below' → $<$
• 'between a and b' → usually $a<x<b$ (read the question to see if the ends are included)

Typical models:

• Average marks: (sum of marks) ÷ (number of tests) ≥ target.
• Consecutive numbers: x and x + 1 (or x and x + 2 for odd/even), then list pairs that fit.
• Temperature: substitute the conversion formula into a double inequality.
• Mixtures: (amount of acid) compared with (percentage × total volume), giving two inequalities.

Always state the final answer in words with units, and respect the context (lengths positive, counts whole).

Worked example

Asha scored 62 and 78 in two unit tests. Find the minimum marks she needs in the third test to average at least 75. [2 marks]

Try it first, then show the solutionSolution

1. Let x be the third mark: $\frac{62+78+x}{3}\ge 75$
2. $140+x\ge 225\Rightarrow x\ge 85$

Answer: At least 85 marks.

Maths in real life: Linear Inequalities

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.

Topics in this chapter

• Solving Linear Inequalities: Rules, Fractions and Double Inequalities
• Systems of Inequalities, Modulus Inequalities and Word Problems

Linear Inequalities formula sheet

Adding a constant

$a<b\Rightarrow a+c<b+c\text{ for every real }c$

Multiplying by a positive number

$a<b,k>0\Rightarrow ka<kb$

Multiplying by a negative number (sign flips)

$a<b,k<0\Rightarrow ka>kb$

Double inequality

$p<ax+b<q,a>0\Rightarrow \frac{p-b}{a}<x<\frac{q-b}{a}$

Inequalities to intervals

$x<a⟺(-\infty ,a),x\ge a⟺[a,\infty ),a<x\le b⟺(a,b]$

Modulus as a double inequality

$|x-c|<a⟺c-a<x<c+a(a>0)$

How the school exam asks Linear Inequalities

The question patterns to expect, and the method that works for each.

Tricks to remember Linear Inequalities

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

• Negative flips the sign

  Multiply or divide by a negative: turn the sign round. Like looking in a mirror: left becomes right.

  Use it when: Every time you divide by a negative coefficient.

• Open for strict, closed for equal

  < and > leave the door open (open circle, round bracket); ≤ and ≥ shut it (filled dot, square bracket).

  Use it when: Number lines and intervals.

• At least = no less = ≥

  'At least 60' means 60 or more: ≥ 60. 'At most 60' means 60 or less: ≤ 60.

  Use it when: Word problems.

• Three-part sandwich

  Whatever you do to the middle, do to the bread on both sides.

  Use it when: Double inequalities.

• System = overlap

  Shade both answers on one number line and keep only the overlap.

  Use it when: Systems of inequalities.

Linear Inequalities: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 1 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Linear Inequalities

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

• ✗Mistake: Not reversing the sign after dividing by a negative

  Solving it exactly like an equation.

  Wrong working

  $-2x>8\Rightarrow x>-4$

  Right working

  $-2x>8\Rightarrow x<-4$

  ✓Do this: Circle any negative divisor and flip the sign in the same line.

  Usually costs: Usually the whole answer.

• ✗Mistake: Giving real solutions when x is restricted to N or Z

  Ignoring the 'when x is a natural number' part.

  ✓Do this: List the allowed values that satisfy the inequality.

  Usually costs: 1 mark.

• ✗Mistake: Wrong circle on the number line

  Not linking strict and slack signs to open and filled circles.

  ✓Do this: Open circle for < or >; filled dot for ≤ or ≥.

  Usually costs: 1 mark.

• ✗Mistake: Reversing only one sign in a double inequality

  Dividing by a negative and fixing just one side.

  Wrong working

  $-38\le -2x<2\Rightarrow 19\le x>-1$

  Right working

  $19\ge x>-1$, i.e. $-1<x\le 19$

  ✓Do this: Reverse both signs, then rewrite from smallest to largest.

  Usually costs: 1-2 marks.

• ✗Mistake: Using 'more than' as ≥ in word problems

  Careless translation.

  ✓Do this: 'More than' is strict (>); 'at least' is ≥.

  Usually costs: 1 mark.

• ✗Mistake: Multiplying both sides by x

  Trying to clear a variable denominator.

  ✓Do this: The sign of x is unknown, so this can reverse the inequality; such questions are not linear and are not asked in this chapter.

  Usually costs: Whole question.

NCERT exercise guide: Linear Inequalities

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Linear Inequalities

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

1. Q1 · 1 mark · MCQExam level

   If $x<5$, then:
   (a) $-x<-5$
   (b) $-x\le -5$
   (c) $-x>-5$
   (d) $-x\ge -5$

   Hint

   Multiply by −1.

   Answer

   (c)

2. Q2 · 1 mark · MCQExam level

   The solution of $1\le 2x-3\le 5$ is:
   (a) [2, 4]
   (b) (2, 4)
   (c) [2, 4)
   (d) (2, 4]

   Hint

   Add 3, divide by 2.

   Answer

   (a)

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): If $\frac{x}{2}+\frac{x}{3}<10$, then $x<12$.
   Reason (R): Multiplying both sides of an inequality by a positive number does not change its direction.
   (a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

   Hint

   Multiply by 6.

   Answer

   (a): $3x+2x<60\Rightarrow x<12$.

4. Q4 · 1 mark · Very short answerExam level

   Solve $4x-5<9$ when (i) x is an integer (ii) x is a real number.

   Hint

   $x<\frac{7}{2}$.

   Answer

   (i) {…, 1, 2, 3} (ii) $(-\infty ,\frac{7}{2})$

5. Q5 · 2 marks · Very short answerHarder exam level

   Solve $4(3-x)\ge 3(2-x)$ for real x.

   Hint

   Expand first.

   Answer

   $12-4x\ge 6-3x\Rightarrow 6\ge x$; $(-\infty ,6]$.

6. Q6 · 2 marks · Very short answerHarder exam level

   Solve $3\le -3(x-5)<9$.

   Hint

   Divide all parts by −3 first.

   Answer

   $-1\ge x-5>-3\Rightarrow 4\ge x>2$, i.e. $(2,4]$.

7. Q7 · 2 marks · Very short answerHarder exam level

   Find all pairs of consecutive odd natural numbers, both larger than 7, whose sum is less than 30.

   Hint

   x and x + 2, x odd, x > 7.

   Answer

   $2x+2<30\Rightarrow x<14$; x = 9, 11, 13: (9, 11), (11, 13), (13, 15).

8. Q8 · 3 marks · Short answerHarder exam level

   Solve the system $3x-5>2(x-4)$ and $7-x>11-3x$, and show the solution on a number line.

   Hint

   Solve each separately.

   Answer

   First: $3x-5>2x-8\Rightarrow x>-3$. Second: $2x>4\Rightarrow x>2$. Common: $x>2$, i.e. $(2,\infty )$; open circle at 2, shaded to the right.

9. Q9 · 3 marks · Short answerHarder exam level

   How many litres of a 3% salt solution must be added to 400 litres of a 9% salt solution so that the mixture is more than 5% but less than 7% salt?

   Hint

   Salt: 0.03x + 36.

   Answer

   $0.03x+36>0.05(400+x)\Rightarrow 16>0.02x\Rightarrow x<800$; $0.03x+36<0.07(400+x)\Rightarrow 8<0.04x\Rightarrow x>200$. So $200<x<800$ litres.

10. Q10 · 2 marks · Very short answerHOTS / competency

    A student says: 'From $\frac{1}{x}<2$ we get $1<2x$, so $x>\frac{1}{2}$.' Show by an example that this is wrong and explain the error.

    Hint

    Try x = −1.

    Answer

    x = −1 satisfies $\frac{1}{x}=-1<2$ but not $x>\frac{1}{2}$. Multiplying by x is only safe if x > 0; for negative x the sign must reverse. So the step lost all negative solutions.

11. Q11 · 2 marks · Very short answerHOTS / competency

    The cost of producing x units is $C=26000+30x$ and the revenue is $R=43x$ (in ₹). How many units must be sold to make a profit?

    Hint

    Profit means R > C.

    Answer

    $43x>26000+30x\Rightarrow 13x>26000\Rightarrow x>2000$: more than 2000 units.

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. 1.What happens to an inequality when both sides are multiplied by −3?Show answer

   The sign reverses.

2. 2.Write $x\ge 4$ as an interval.Show answer

   $[4,\infty )$.

3. 3.Solve $-9x>20$ for natural x.Show answer

   No solution ($x<-\frac{20}{9}$).

4. 4.Open or filled circle for $x<3$?Show answer

   Open circle at 3.

5. 5.Translate 'at most 40'.Show answer

   $\le 40$.

6. 6.Solve $5\le 2x+1\le 9$.Show answer

   $2\le x\le 4$.

7. 7.What is the solution of a system?Show answer

   The values common to all the inequalities (the intersection).

8. 8.Is graphing $2x+3y\le 6$ in the 2026-27 annual exam?Show answer

   No, inequalities in two variables are assessed only formatively.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you state both rules for solving inequalities?
• ☐Can you explain why the sign reverses for a negative multiplier?
• ☐Can you solve an inequality for natural numbers and integers?
• ☐Can you clear fractions without changing the sign?
• ☐Can you show a solution on the number line with the right circles?
• ☐Can you write solutions as intervals?
• ☐Can you solve a double inequality with a negative coefficient?
• ☐Can you solve a system and show the common part?
• ☐Can you translate 'at least', 'more than' and 'between'?
• ☐Can you set up a mixture problem as two inequalities?

Linear Inequalities: frequently asked questions