Linear Inequalities: Class 11 Maths · Chapter 5

Mathematics · Companion notes, simulations and practice · Official NCERT chapter PDF ↗

Published 18 Oct 2026 · Updated 18 Oct 2026

50 min
28 practice questions
14 MCQs

What you will learn in Linear Inequalities

Recognise inequalities in everyday statements and write them with <, >, ≤ or ≥.
Classify an inequality as numerical, literal or double, strict or slack, and linear or not.
Find the solutions and the solution set of a linear inequality in one variable over N, Z or R.
Solve linear inequalities with the two rules, reversing the sign when multiplying or dividing by a negative number.
Show a solution set on a number line and write it in interval notation.
Solve double inequalities and systems of two linear inequalities by finding the common solutions.
Model word problems on averages, consecutive numbers, lengths, temperature ranges and mixtures as linear inequalities and interpret the answers.

Original companion text aligned to NCERT Mathematics Class 11, Chapter 5, pp. 89–99. Not affiliated with or endorsed by NCERT.

5.1 Introduction

The lift in a housing society carries a plate: maximum 8 persons or 544 kg. A film certificate says the viewer must be at least 18. The vaccine fridge at a primary health centre must stay between 2 °C and 8 °C. None of these fixes a single value. Each sets a limit, and many values respect it.

Statements of this kind use four signs: $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to) and $\geq$ (greater than or equal to). A statement built with one of them is an inequality (असमिका). Economists use inequalities for budgets, engineers for safety margins and tolerances, doctors for safe dose ranges and statisticians for error bounds.

An equation usually pins the unknown down to one value; an inequality usually allows a whole stretch of values (Fig. 5.1). In this chapter you will learn to find that whole stretch for a linear inequality in one variable, and to draw it on a number line.

2/2Right pan heavier: $2 x + 1 < 7$ holds for every $x < 3$ .

Fig. 5.1An equation balances for one value; an inequality holds for a whole stretch of values.

The left pan of a balance holds two boxes marked x and a weight marked 1; the right pan holds a weight marked 7. In the first step the beam is level, the statement reads 2x + 1 = 7, and the number line below has a single filled dot at 3. In the second step the beam tilts so the right pan is lower, the statement reads 2x + 1 < 7, and the number line shows a hollow circle at 3 with a thick ray running left from it, because every number below 3 works.

5.2 Inequalities

Start with a purchase. Your cricket team needs bottles of water for a match. A bottle costs ₹20 and the team kitty holds ₹150. If you buy $x$ bottles you spend $20 x$ rupees, and that cannot be more than the money in the kitty:

20 x \leq 150

(5.1)

No equals sign forces one value here. In fact you can never spend the whole ₹150, because 150 is not a multiple of 20: seven bottles leave ₹10, and an eighth bottle is ₹10 too many (Fig. 5.2).

3/4 $x = 7$ costs ₹140, leaving ₹10.

Fig. 5.2Whole bottles at ₹20 each: seven fit into ₹150, eight do not.

A long empty frame stands for a budget of ₹150, with a scale marked ₹0, ₹50, ₹100 and ₹150 underneath. In the first step it is empty and the statement 20x ≤ 150 is shown. In the second step five bottle blocks fill ₹100 of the frame and the label says it fits the budget. In the third step seven blocks fill ₹140 and a label notes ₹10 left over. In the fourth step an eighth block, drawn dashed, sticks out past the end of the frame, with labels saying ₹10 over and over budget, since 20 × 8 = 160 is more than 150.

Now let two quantities vary. Notebooks cost ₹50 each and pens ₹10 each, and you have ₹200 for $x$ notebooks and $y$ pens. The bill is $50 x + 10 y$ rupees, so

50 x + 10 y \leq 200

(5.2)

Some baskets use every rupee: three notebooks and five pens cost exactly ₹200. Others leave change: two notebooks and four pens cost ₹140. So the one sign $\leq$ hides two separate statements, the strict inequality $50 x + 10 y < 200$ for baskets that leave change and the equation $50 x + 10 y = 200$ for baskets that use it all.

Definition

Inequality

A statement in which one of the signs $<$ , $>$ , $\leq$ or $\geq$ stands between two quantities, each a real number or an algebraic expression.

Example $7 > - 2$ and $3 x - 1 \leq 8$ are both inequalities.

A comparison between numbers alone, such as $π > 3$ or $- 5 < - 2$ , is a numerical inequality: it is simply true or false. One that contains a variable is a literal inequality. On a road with a limit of 60 km/h, a speed $v$ must obey $v \leq 60$ , and whether that holds depends on the value of $v$ . A double inequality traps a quantity between two bounds. The mark band $40 \leq m < 60$ is read " $m$ is at least 40 and less than 60", so 40 belongs to the band and 60 does not.

The signs $<$ and $>$ make a strict inequality, which shuts out the boundary value; the signs $\leq$ and $\geq$ make a slack inequality, which lets it in. An inequality is linear when neither side has a variable raised to a power above 1 or multiplied by another variable. Table 5.1 sorts some examples.

Table 5.1Kinds of inequality

Example	What kind?
$3 x - 7 < 0$	linear, one variable, strict
$5 - 2 t \geq 0$	linear, one variable, slack
$2 x + 3 y > 6$	linear, two variables, strict
$4 x - y \leq 10$	linear, two variables, slack
$x^{2} \leq 9$	quadratic, one variable, slack
$1 < 2 p \leq 9$	linear, one variable, double

Definition

Linear inequality

In one variable $x$ , an expression $a x + b$ with $a \neq 0$ is compared with 0 by one of the four signs. In two variables, $a x + b y + c$ is compared with 0, where $a \neq 0$ and $b \neq 0$ .

Example $5 - 2 t \geq 0$ is linear in $t$ ; $x^{2} \leq 9$ is not linear.

The row with $x^{2}$ is a quadratic inequality, outside this chapter. From here on we work with linear inequalities in one variable. Inequalities in two variables come back in Class 12, where their solutions fill whole regions of the plane in linear programming.

5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

Go back to the bottles, $20 x \leq 150$ . Here $x$ counts bottles, so it can only be 0, 1, 2 and so on. With so few candidates, the honest first move is to try them (Table 5.2).

Table 5.2Testing whole numbers in

20 x \leq 150

$x$	$20 x$	Is $20 x \leq 150$ ?
0	0	true
1	20	true
2	40	true
3	60	true
4	80	true
5	100	true
6	120	true
7	140	true
8	160	false

Every value from 0 to 7 passes, and 8 fails; larger values fail too, because $20 x$ only grows. Each value that passes the test is called a solution (हल) of the inequality.

Definition

Solution set

The set of all solutions of an inequality, taken from the values the variable is allowed to have.

ExampleFor whole numbers of bottles, $20 x \leq 150$ has the solution set ${0, 1, 2, \dots, 7} .$

Trying values works only because the answers here are few and whole. If $x$ could be any real number, no table could ever list the solutions, so we need a method. With equations you already have one: add or subtract the same number on both sides, or multiply or divide both sides by the same non-zero number, until $x$ stands alone. Inequalities allow nearly the same moves, with one twist.

The two rules

LawRule 1 (adding or subtracting)

Shifting both sides by the same amount (adding one number to each side, or taking one number away from each side) leaves the inequality pointing the same way: if $a < b$ , then $a + c < b + c$ for every real number $c$ .

LawRule 2 (multiplying or dividing)

Scaling both sides by a positive number (multiplying or dividing) preserves the inequality; scaling by a negative number turns it round. In symbols, if $a < b$ , then $a c < b c$ for $c > 0$ , but $a c > b c$ for $c < 0$ .

Rule 2 is where most mistakes happen, and a number line shows why it must be so (Fig. 5.3). Adding, subtracting or multiplying by a positive number slides or stretches the line without turning it round. Multiplying by $- 1$ reflects every point in 0, so whichever point was on the left lands on the right. The same reversal applies to $\leq$ and $\geq$ : dividing $- 2 x \geq 6$ by $- 2$ gives $x \leq - 3$ .

4/4Rule 2, negative: each point is reflected in 0 and the order flips: $6 > - 3$ .

Fig. 5.3Shifting and stretching keep the order of two numbers; multiplying by a negative number reverses it.

A number line from −8 to 8 carries a round point a and a square point b. In the first step a is at 2 and b at 5, so 2 < 5. In the second step both slide 4 units left to −2 and 1, still with a on the left, and the label says order kept. In the third step both are multiplied by 3 and move to −6 and 3, again with a on the left and order kept. In the fourth step both are multiplied by −1: a jumps to 6 and b to −3, a dashed mirror line at 0 appears, and the label says order reversed, because now a is on the right.

From now on each step names the rule it uses. Rule 1 is what lets you move a term across by changing its sign, exactly as you do with equations.

Example 5.1NCERT Example 1 · p. 92

Natural numbers or integers

Find the solution set of $12 x \leq 75$ (i) in the natural numbers, (ii) in the integers.

Try it yourself first. Then reveal the solution one step at a time.

Example 5.2NCERT Example 2 · p. 92

Integers or real numbers

Find every solution of $7 x + 2 \leq 4 x + 14$ (i) among the integers, (ii) among the real numbers.

Try it yourself first. Then reveal the solution one step at a time.

Over the real numbers the solutions fill a whole interval (अंतराल), written in the notation of Chapter 1. A round bracket leaves an end point out and a square bracket keeps it in; $\infty$ always takes a round bracket because it is not a number. Unless a question says otherwise, from now on $x$ is a real number.

Example 5.3NCERT Example 3 · p. 93

A negative coefficient

Solve $3 x - 5 > 7 x + 3.$

Try it yourself first. Then reveal the solution one step at a time.

Example 5.4NCERT Example 4 · p. 93

Clearing fractions

Solve $\frac{2 - x}{3} \geq \frac{x + 4}{2} .$

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 4

Solve

- 3 x \leq 12

x \leq - 4

x \leq 4

x \geq 4

x \geq - 4

Show answer

Answer (D) $x \geq - 4$

Dividing by

- 3

reverses the sign, so

x \geq - 4

. Keeping the sign (option A) is the most common slip.

Showing solutions on a number line

A sketch of the solution set often says more than the inequality itself. On a number line, mark the boundary number with a circle. Then draw a thick line over the part where the solutions lie, with an arrowhead to show it carries on for ever. The circle tells you whether the boundary belongs to the set (Fig. 5.4):

a hollow (open) circle for $<$ and $>$ , because the boundary is not a solution;
a filled (solid) circle for $\leq$ and $\geq$ , because the boundary is a solution;
shading to the left for $x < a$ and $x \leq a$ , and to the right for $x > a$ and $x \geq a$ .

2/4Filled circle at 2, shaded left: $(- \infty, 2] .$

Fig. 5.4Hollow circle: the boundary is left out. Filled circle: it is included.NCERT Figs 5.1, 5.2 · p. 93

A number line from −5 to 5. In the first step a hollow circle at 2 has a thick ray running left, with the label x < 2, the interval from minus infinity to 2 with a round bracket, and a note that 2 is left out. In the second step the same ray has a filled circle at 2, labelled x ≤ 2 with a square bracket, and the note says 2 is included. In the third step a hollow circle at −3 has a ray running right, labelled x > −3. In the fourth step that circle is filled, labelled x ≥ −3 with a square bracket at −3.

Example 5.5NCERT Example 5 · p. 93

Solve, then draw

Solve $6 x + 5 > 2 x - 7$ and show the solutions on a number line.

Try it yourself first. Then reveal the solution one step at a time.

Example 5.6NCERT Example 6 · p. 93

Fractions, then a graph

Solve $\frac{3 x + 1}{2} \leq \frac{x + 9}{4} + 2$ and show the solutions on a number line.

Try it yourself first. Then reveal the solution one step at a time.

The lab below solves any inequality of the form $a x + b < c x + d$ (or with $\leq$ , $>$ , $\geq$ ) one rule at a time. Watch what happens to the sign when the coefficient you divide by turns negative, then switch to integers or natural numbers and see the shaded ray break into separate dots.

InteractiveInequality lab

Four sliders set the integers a, b, c and d from −9 to 9, buttons choose the sign <, ≤, > or ≥, and three buttons choose whether x is a real number, an integer or a natural number. The lab writes the inequality and lists the solving steps, naming Rule 1 or Rule 2 and highlighting the step where dividing by a negative number reverses the sign; when the x-terms cancel it says whether every value or no value works. A number line from −10 to 10 shades the solution ray with a hollow or filled circle (or marks dots for integer and natural-number solutions) and states the solution set. A test slider places a marker at any x and reports both sides and whether the statement is true.

Fig. 5.5Build

a x + b ? c x + d,

follow the steps, and test values of

x

on the number line.

Word problems

Phrases such as at least, at most, no more than and exceeds are inequality signs in disguise. At least means $\geq$ , at most and no more than mean $\leq$ , and exceeds means $>$ . Name the unknown, write the condition and solve it. Then check that the answer makes sense in the story: people and bottles come in whole numbers, and a mark cannot be more than the maximum.

Example 5.7NCERT Example 7 · p. 94

Holding an average

In a school tournament Ishita scored 42, 35 and 61 runs in her first three innings. How many runs must she score in the fourth innings to average at least 50 runs per innings?

Try it yourself first. Then reveal the solution one step at a time.

Example 5.8NCERT Example 8 · p. 94

Consecutive multiples of 5

Two consecutive multiples of 5 are each greater than 20, and together they add up to less than 75. Which pairs are possible?

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 3

A number line shows a filled circle at

- 1

and shading to the right. Which inequality is drawn?

x > - 1

x \leq - 1

x \geq - 1

x < - 1

Show answer

Answer (C) $x \geq - 1$

Filled circle:

- 1

is included. Shading right: larger values. Together,

x \geq - 1

. Option A would need a hollow circle.

Practice · 14 questions

Practice 5.1: questions with answers

1
Short answerEasyNCERT Exercise 5.1, Q1
Find all solutions of $14 x \leq 90$
1. among the natural numbers;
  Show solution
  Answer ${1, 2, \dots, 6}$
  $x \leq \frac{90}{14} = \frac{45}{7},$ about 6.43. The natural numbers up to 6.43 are 1 to 6.
2. among the integers.
  Show solution
  Answer ${\dots, 4, 5, 6}$
  The bound is the same, $x \leq \frac{45}{7} .$ Every integer up to 6 qualifies, with no lower end.
2
Short answerEasyNCERT Exercise 5.1, Q2
List the solutions of $- 15 x \geq 48$
1. in $𝐍;$
  Show solution
  Answernone: $\emptyset$
  Divide by $- 15$ and reverse the sign: $x \leq - \frac{48}{15} = - 3.2.$ No natural number is negative, so the solution set is empty.
2. in $𝐙 .$
  Show solution
  Answer ${\dots, - 5, - 4}$
  The integers that do not exceed $- 3.2$ are $- 4, - 5, - 6, \dots$
3
Short answerEasyNCERT Exercise 5.1, Q3
Solve $6 x - 5 < 13$ over
1. the integers;
  Show solution
  Answer ${\dots, 0, 1, 2}$
  $6 x < 18$ , so $x < 3$ . The integers below 3 are $2, 1, 0, - 1, \dots;$ 3 itself is excluded.
2. the real numbers.
  Show solution
  Answer $(- \infty, 3)$
  Every real number less than 3, which is the interval $(- \infty, 3) .$
4
Short answerEasyNCERT Exercise 5.1, Q4
Which values satisfy $2 x + 11 \geq 4$ ?
1. Integer values.
  Show solution
  Answer ${- 3, - 2, - 1, \dots}$
  $2 x \geq - 7$ , so $x \geq - 3.5$ . The smallest integer that is at least $- 3.5$ is $- 3$ .
2. Real values.
  Show solution
  Answer $[- \frac{7}{2}, \infty)$
  Every real number from $- 3.5$ upwards, with $- 3.5$ included.
5
Short answerEasyNCERT Exercise 5.1, Q5–Q8
Solve for real $x$ .
1. $7 x + 2 < 9 x - 6$
  Show solution
  Answer $(4, \infty)$
  $- 2 x < - 8$ . Divide by $- 2$ and reverse: $x > 4$ .
2. $5 x + 4 > 8 x - 11$
  Show solution
  Answer $(- \infty, 5)$
  $- 3 x > - 15$ , so $x < 5$ .
3. $4 (x + 2) \geq 3 (x - 1)$
  Show solution
  Answer $[- 11, \infty)$
  $4 x + 8 \geq 3 x - 3,$ so $x \geq - 11$ .
4. $5 (1 - x) < 2 (4 - x)$
  Show solution
  Answer $(- 1, \infty)$
  $5 - 5 x < 8 - 2 x,$ so $- 3 x < 3$ and $x > - 1$ .
6
Short answerMediumNCERT Exercise 5.1, Q9–Q12
Solve for real $x$ .
1. $x - \frac{x}{3} + \frac{x}{4} < 22$
  Show solution
  Answer $(- \infty, 24)$
  Multiply by 12: $12 x - 4 x + 3 x < 264,$ so $11 x < 264$ and $x < 24$ .
2. $\frac{x}{5} < \frac{x}{2} + 3$
  Show solution
  Answer $(- 10, \infty)$
  Multiply by 10: $2 x < 5 x + 30,$ so $- 3 x < 30$ and $x > - 10$ .
3. $\frac{2 (x + 1)}{3} \geq \frac{3 (x - 4)}{4}$
  Show solution
  Answer $(- \infty, 44]$
  Multiply by 12: $8 (x + 1) \geq 9 (x - 4),$ so $8 x + 8 \geq 9 x - 36.$ Then $- x \geq - 44$ and $x \leq 44$ .
4. $\frac{1}{3} (\frac{x}{2} + 5) < \frac{1}{4} (x - 2)$
  Show solution
  Answer $(26, \infty)$
  Multiply by 12: $4 (\frac{x}{2} + 5) < 3 (x - 2),$ so $2 x + 20 < 3 x - 6$ and $x > 26$ .
7
Short answerMediumNCERT Exercise 5.1, Q13–Q16
Solve for real $x$ .
1. $3 (x - 2) + 10 \leq 5 (x + 1)$
  Show solution
  Answer $[- \frac{1}{2}, \infty)$
  $3 x + 4 \leq 5 x + 5,$ so $- 2 x \leq 1$ and $x \geq - \frac{1}{2} .$
2. $25 - (2 x - 3) < 7 x - 4 (x + 2)$
  Show solution
  Answer $(\frac{36}{5}, \infty)$
  The left side is $28 - 2 x$ and the right side is $3 x - 8$ . So $- 5 x < - 36$ and $x > \frac{36}{5} .$
3. $\frac{x}{3} > \frac{3 x + 1}{4} - \frac{x - 2}{6}$
  Show solution
  Answer $(- \infty, - \frac{7}{3})$
  Multiply by 12: $4 x > 3 (3 x + 1) - 2 (x - 2),$ and the right side is $7 x + 7$ . So $- 3 x > 7$ and $x < - \frac{7}{3} .$
4. $\frac{3 x}{5} \leq \frac{x + 1}{2} - \frac{1 - x}{4}$
  Show solution
  Answer $[- \frac{5}{3}, \infty)$
  Multiply by 20: $12 x \leq 10 (x + 1) - 5 (1 - x),$ that is, $12 x \leq 15 x + 5.$ So $- 3 x \leq 5$ and $x \geq - \frac{5}{3} .$
8
DiagramMediumNCERT Exercise 5.1, Q17–Q20
Solve each inequality and describe its graph on a number line.
1. $4 x + 1 < 3 x + 5$
  Show solution
  Answer $x < 4$
  $x < 4$ : a hollow circle at 4, shaded to the left.
2. $6 x - 2 > 4 x - 8$
  Show solution
  Answer $x > - 3$
  $2 x > - 6$ , so $x > - 3$ : a hollow circle at $- 3$ , shaded to the right.
3. $4 (2 - x) \leq 3 (x + 5)$
  Show solution
  Answer $x \geq - 1$
  $8 - 4 x \leq 3 x + 15,$ so $- 7 x \leq 7$ and $x \geq - 1$ : a filled circle at $- 1$ , shaded to the right.
4. $\frac{x}{3} \geq \frac{2 x + 1}{2} - \frac{x + 3}{4}$
  Show solution
  Answer $x \leq \frac{3}{5}$
  Multiply by 12: $4 x \geq 6 (2 x + 1) - 3 (x + 3),$ and the right side is $9 x - 3$ . So $- 5 x \geq - 3$ and $x \leq \frac{3}{5}$ : a filled circle at $\frac{3}{5}$ , shaded to the left.
NumericalEasyNCERT Exercise 5.1, Q21
Kavya's monthly tests are each marked out of 50. She scored 34 and 41 in the first two. What is the least she must score in the third test to average at least 38?
Show solution
Answer39 marks
$\frac{34 + 41 + x}{3} \geq 38$ gives $75 + x \geq 114,$ so $x \geq 39$ . The test is out of 50, so any score from 39 to 50 will do.
NumericalEasyNCERT Exercise 5.1, Q22
A cycling club gives a badge for an average of at least 25 km per ride over six rides. Arjun's first five rides were 22, 27, 24, 21 and 26 km. How long must his sixth ride be to earn the badge?
km
Show solution
AnswerAt least 30 km
The five rides total 120 km. Then $\frac{120 + x}{6} \geq 25$ gives $120 + x \geq 150,$ so $x \geq 30$ .
Short answerMediumNCERT Exercise 5.1, Q23
Two consecutive odd natural numbers are both less than 18 and add up to more than 20. List every such pair.
Show solution
Answer(11, 13), (13, 15), (15, 17)
Both less than 18 means $x + 2 < 18$ , so $x < 16$ . The sum condition $2 x + 2 > 20$ gives $x > 9$ . The odd numbers with $9 < x < 16$ are 11, 13 and 15.
Short answerMediumNCERT Exercise 5.1, Q24
Pairs of consecutive multiples of 4 are wanted in which both numbers exceed 10 and the total stays below 50. Find them all.
Show solution
Answer(12, 16), (16, 20), (20, 24)
With $x$ and $x + 4$ : $x > 10$ and $2 x + 4 < 50$ , so $10 < x < 23$ . The multiples of 4 in this range are 12, 16 and 20.
NumericalMediumNCERT Exercise 5.1, Q25
In a triangle, one side is 6 cm longer than the shortest side, and the longest side is 1 cm more than twice the shortest. The perimeter is at least 55 cm. Find the least possible length of the shortest side.
cm
Show solution
Answer12 cm
$x + (x + 6) + (2 x + 1) \geq 55$ gives $4 x + 7 \geq 55,$ so $x \geq 12$ . A triangle really exists: the two shorter sides add to $2 x + 6$ , which always beats $2 x + 1$ . For $x = 12$ the sides are 12, 18 and 25 cm.
Short answerHardNCERT Exercise 5.1, Q26
A tailor cuts three pieces from a 150 cm ribbon (some ribbon may be left over). The second piece is 10 cm longer than the shortest, and the third is three times the shortest. The third piece must be at least 20 cm longer than the second. What lengths are possible for the shortest piece?
Show solution
Answer15 cm to 28 cm
Let the shortest piece be $x$ cm. Fitting: $x + (x + 10) + 3 x \leq 150,$ so $5 x \leq 140$ and $x \leq 28$ . The gap: $3 x \geq (x + 10) + 20,$ so $2 x \geq 30$ and $x \geq 15$ . Hence $15 \leq x \leq 28.$

Miscellaneous Examples

The last examples combine the rules. A double inequality such as $p < a x + b \leq q$ is two inequalities in one line, and you may apply a rule to all three parts at once, provided you apply it to every part. Several inequalities that must hold together form a system of inequalities; its solution set is the common part (the intersection) of the separate solution sets.

Example 5.9NCERT Example 9 · p. 96

Working on three parts

Solve $- 7 < 4 x + 5 \leq 13.$

Try it yourself first. Then reveal the solution one step at a time.

Example 5.10NCERT Example 10 · p. 96

A double inequality that flips

Solve $- 4 \leq \frac{1 - 3 x}{2} < 5.$

Try it yourself first. Then reveal the solution one step at a time.

Follow the end points in Fig. 5.6. The closed end at $- 9$ becomes the closed end at 3, and the open end at 9 becomes the open end at $- 3$ . Dividing by a negative number swaps the two ends, and each end keeps its own type.

4/4 $x$ lies in $(- 3, 3]$ : the closed end moves to 3, the open end to $- 3$ .

Fig. 5.6Dividing by a negative number swaps the ends of the interval; each end keeps its own type.

A number line from −10 to 10. In the first step a segment from −4 (filled end) to 5 (hollow end) shows the values of the middle part (1 − 3x)/2. In the second step, after multiplying by 2, the segment runs from −8 (filled) to 10 (hollow). In the third step, after subtracting 1, it runs from −9 (filled) to 9 (hollow). In the fourth step, after dividing by −3, the segment for x runs from −3 (hollow) to 3 (filled); the old segment is shown dashed and two crossing arrows carry −9 to 3 and 9 to −3, showing that the ends swap sides and keep their types.

Example 5.11NCERT Example 11 · p. 96

A system of two inequalities

Solve the system $4 x - 3 \geq x - 6$ and $9 - 2 x > 1,$ and show the solutions on a number line.

Try it yourself first. Then reveal the solution one step at a time.

3/3Both hold on $[- 1, 4)$ .

Fig. 5.7The solution of a system is the overlap of the separate solution sets.NCERT Fig. 5.3 · p. 97

Three number lines from −3 to 6 are stacked. In the first step the top line shows x ≥ −1 as a filled circle at −1 with a ray to the right. In the second step the middle line adds x < 4 as a hollow circle at 4 with a dashed ray to the left. In the third step dashed guides drop from −1 and 4 to the bottom line, which shows only the overlap: a segment from a filled circle at −1 to a hollow circle at 4, labelled −1 ≤ x < 4.

Example 5.12NCERT Example 12 · p. 97

A vaccine fridge

Many vaccines must be stored from 2 °C to 8 °C. What is this range in degrees Fahrenheit? Use $F = \frac{9}{5} C + 32.$

Try it yourself first. Then reveal the solution one step at a time.

3/3The same range is $35.6 \leq F \leq 46.4.$

Fig. 5.8The same safe range, measured on two scales.

Two parallel scales: degrees Celsius from 0 to 10 on top and degrees Fahrenheit from 32 to 50 below, lined up so that the same temperature sits at the same horizontal position. In the first step a band on the Celsius scale marks 2 to 8. In the second step dashed arrows drop straight down from 2 and 8 to the Fahrenheit scale, with the formula F = 9C/5 + 32 between the scales. In the third step the matching band on the Fahrenheit scale is shaded and labelled 35.6 and 46.4.

Example 5.13NCERT Example 13 · p. 97

Mixing two fertiliser solutions

A farmer's tank holds 500 litres of liquid fertiliser with 10 % nutrient. She will top it up with a stronger fertiliser that is 40 % nutrient, aiming for a mix whose nutrient content lies strictly between 16 % and 20 %. What amounts of the stronger fertiliser will do?

Try it yourself first. Then reveal the solution one step at a time.

Fig. 5.9 shows why the answer is a window. As more of the strong solution goes in, the strength of the mix climbs from 10 % towards 40 %, crossing 16 % at 125 litres and 20 % at 250 litres.

3/3The curve is inside the band for $125 < x < 250.$

Fig. 5.9The mix is between 16% and 20% exactly when between 125 and 250 litres are added.

A graph with litres of 40% solution added, from 0 to 400, across and the strength of the mix, from 10% to 24%, up. In the first step a curve starts at 10% when nothing is added and rises more and more slowly, reaching about 23% at 400 litres. In the second step a shaded band between the 16% and 20% lines appears. In the third step dashed lines drop from where the curve enters and leaves the band to the horizontal axis, and the stretch between 125 and 250 litres is marked with hollow end circles, because both conditions are strict.

Quick check

1 of 3

Solve

1 \leq 3 - 2 x < 7.

(- 2, 1]

[- 2, 1)

(- 1, 2]

[1, 3)

Show answer

Answer (A) $(- 2, 1]$

- 2 \leq - 2 x < 4;

dividing by

- 2

gives

1 \geq x > - 2,

(- 2, 1]

. Option B forgets that the ends swap when the signs reverse.

Recap

Linear Inequalities: key points and summary

An inequality compares two numbers or expressions with $<$ , $>$ , $\leq$ or $\geq$ ; at least means $\geq$ and at most means $\leq$ (Section 5.1).
Inequalities are numerical or literal, single or double, strict ( $<$ , $>$ ) or slack ( $\leq$ , $\geq$ ), and linear or not (Section 5.2).
A solution makes the inequality true; the solution set collects all of them and depends on whether $x$ ranges over $𝐍$ , $𝐙$ or $𝐑$ (Section 5.3).
Rule 1: adding or subtracting the same number on both sides keeps the sign (Law).
Rule 2: multiplying or dividing by a positive number keeps the sign; by a negative number it reverses the sign (Law).
Never multiply or divide by an expression whose sign is unknown, or by zero (Think about it).
On a number line use a hollow circle for $<$ and $>$ , a filled circle for $\leq$ and $\geq$ , and shade the side where the solutions lie (Showing solutions on a number line).
Over $𝐑$ the answers are intervals such as $(- \infty, a),$ $(- \infty, a],$ $(a, \infty)$ and $[a, \infty)$ (Example 5.2).
A double inequality is solved by acting on all three parts; a negative divisor reverses both signs and swaps the ends (Example 5.10).
A system's solution set is the overlap of the separate solution sets, and it may be empty (Example 5.11).
In word problems, check that the answer fits the story: whole numbers, maximum marks, physical limits (Word problems).

Find your NCERT exercise question

Our practice questions are written for ncert.ai and follow the NCERT exercises. Find the one that matches the question in your textbook; its answer is with it.

In your NCERT textbook	Practice question here
NCERT Exercise 5.1, Q1	Practice 5.1 · Q1
NCERT Exercise 5.1, Q2	Practice 5.1 · Q2
NCERT Exercise 5.1, Q3	Practice 5.1 · Q3
NCERT Exercise 5.1, Q4	Practice 5.1 · Q4
NCERT Exercise 5.1, Q5–Q8	Practice 5.1 · Q5
NCERT Exercise 5.1, Q9–Q12	Practice 5.1 · Q6
NCERT Exercise 5.1, Q13–Q16	Practice 5.1 · Q7
NCERT Exercise 5.1, Q17–Q20	Practice 5.1 · Q8
NCERT Exercise 5.1, Q21	Practice 5.1 · Q9
NCERT Exercise 5.1, Q22	Practice 5.1 · Q10
NCERT Exercise 5.1, Q23	Practice 5.1 · Q11
NCERT Exercise 5.1, Q24	Practice 5.1 · Q12
NCERT Exercise 5.1, Q25	Practice 5.1 · Q13
NCERT Exercise 5.1, Q26	Practice 5.1 · Q14
NCERT Miscellaneous Exercise, Q1–Q3	Miscellaneous exercises · Q1
NCERT Miscellaneous Exercise, Q4–Q6	Miscellaneous exercises · Q2
NCERT Miscellaneous Exercise, Q7–Q10	Miscellaneous exercises · Q3
NCERT Miscellaneous Exercise, Q11	Miscellaneous exercises · Q4
NCERT Miscellaneous Exercise, Q12	Miscellaneous exercises · Q5
NCERT Miscellaneous Exercise, Q13	Miscellaneous exercises · Q6
NCERT Miscellaneous Exercise, Q14	Miscellaneous exercises · Q7

Quick check: MCQs on Linear Inequalities

5.2 Inequalities4 questions
The two rules4 questions
Word problems3 questions
Miscellaneous Examples3 questions

Glossary · 11

Key terms (English–Hindi)

double inequality: A statement that places a quantity between two bounds, such as $40 \leq m < 60$ .First seen in the chapter →

inequalityअसमिकाasamika: A statement that compares two numbers or expressions using $<$ , $>$ , $\leq$ or $\geq$ .First seen in the chapter →
intervalअंतरालantaral: An unbroken stretch of the real line, such as $(- 3, 2]$ or $[4, \infty)$ .First seen in the chapter →

linear inequalityरैखिक असमिकाrekhik asamika: An inequality whose two sides are polynomials of degree at most 1 in the variables, such as $3 x - 7 < 0$ or $2 x + 3 y \leq 6$ .First seen in the chapter →
literal inequality: An inequality containing a variable, such as $v \leq 60$ ; its truth depends on the value of the variable.First seen in the chapter →

numerical inequality: An inequality between numbers only, such as $- 5 < - 2$ ; it is simply true or false.First seen in the chapter →

slack inequality: An inequality using $\leq$ or $\geq$ , so the boundary value is allowed.First seen in the chapter →
solutionहलhal: A value of the variable that makes the inequality a true statement.First seen in the chapter →
solution setहल समुच्चयhal samuchchay: The set of all solutions of an inequality from the allowed values of the variable.
strict inequality: An inequality using $<$ or $>$ , so the boundary value is not allowed.First seen in the chapter →
system of inequalities: Two or more inequalities that must hold at the same time; its solutions are the values common to all of them.First seen in the chapter →

Linear Inequalities: Class 11 Maths · Chapter 5

What you will learn in Linear Inequalities

5.1 Introduction

5.2 Inequalities

5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

The two rules

Showing solutions on a number line

Word problems

Practice 5.1: questions with answers

Miscellaneous Examples

Linear Inequalities: key points and summary

Practice questions with answers (NCERT “Miscellaneous exercises”)

Find your NCERT exercise question

Quick check: MCQs on Linear Inequalities

Key terms (English–Hindi)

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