Skip to content
← Mathematics

Mathematics · Class 11 Science

Ch 5Linear Inequalities — Class 11 Mathematics, concept-first.

In earlier classes, you learned how to set up and solve equations — statements of equality such as or — and how to translate real-life word problems into equation form. A natural question arises: can every real-world situation be expressed as an equation? Consider two examples:

94

Q&A

4

Concepts

~4m

Unit weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Linear Inequality Solutions

Imagine you're standing on a number line. You know exactly where the number 5 is. But what if I asked you to stand on "all numbers greater than 5"? You can't stand on all of them at once — they stretch infinitely to the…

Most relevant Q&A

Start with this concept →

In previous exams

How often this chapter’s concepts have been examined — real appearance data, never estimated.

CA Foundation

1 concept · 5 appearances

2026, 2025, 2024

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

5.1

Introduction

In earlier classes, you learned how to set up and solve equations — statements of equality such as or — and how to translate real-life word problems into equation form.

5.2

Inequalities

When we say "Ravi has ₹200 and rice costs ₹30 per packet," we are not writing an equation. If Ravi buys packets, he spends rupees.

5.3

Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation

34 Q

When we solve an inequality, we are looking for all values of the variable that make the inequality true.

+Worked Examplesi8 questions
  1. Example 1Solve $30x < 200$ when (i) $x$ is a natural number, (ii) $x$ is an integer.Free
  2. Example 2Solve $5x - 3 < 3x + 1$ when (i) $x$ is an integer, (ii) $x$ is a real number.Free
  3. Example 3Solve $4x + 3 < 6x + 7$.Free
  4. Example 4Solve $\dfrac{5 - 2x}{3} \le \dfrac{x}{6} - 5$.Preview
  5. Example 5Solve $7x + 3 < 5x + 9$. Show the graph of the solutions on number line.Preview
  6. Example 6Solve $\dfrac{3x - 4}{2} \ge \dfrac{x + 1}{4} - 1$. Show the graph of the solutions on number line.Preview
  7. Example 7The marks obtained by a student of Class XI in first and second terminal examination are 62 and 48, respectively. Find the minimum marks he…Preview
  8. Example 8Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.Preview
+Exercise 5.1i26 questions
  1. Q1Solve $24x < 100$, when (i) $x$ is a natural number. (ii) $x$ is an integer.Free
  2. Q2Solve $-12x > 30$, when (i) $x$ is a natural number. (ii) $x$ is an integer.Free
  3. Q3Solve $5x - 3 < 7$, when (i) $x$ is an integer. (ii) $x$ is a real number.Free
  4. Q4Solve $3x + 8 > 2$, when (i) $x$ is an integer. (ii) $x$ is a real number.Preview
  5. Q5$4x + 3 < 5x + 7$Preview
  6. Q6$3x - 7 > 5x - 1$Preview
  7. Q7$3(x - 1) \le 2(x - 3)$Preview
  8. Q8$3(2 - x) \ge 2(1 - x)$Preview
  9. Q9$x + \dfrac{x}{2} + \dfrac{x}{3} < 11$Preview
  10. Q10$\dfrac{x}{3} > \dfrac{x}{2} + 1$Preview
  11. Q11$\dfrac{3(x - 2)}{5} \le \dfrac{5(2 - x)}{3}$Preview
  12. Q12$\dfrac{1}{2}\left(\dfrac{3x}{5} + 4\right) \ge \dfrac{1}{3}(x - 6)$Preview
  13. Q13$2(2x + 3) - 10 < 6(x - 2)$Preview
  14. Q14$37 - (3x + 5) > 9x - 8(x - 3)$Preview
  15. Q15$\dfrac{x}{4} < \dfrac{5x - 2}{3} - \dfrac{7x - 3}{5}$Preview
  16. Q16$\dfrac{2x - 1}{3} \ge \dfrac{3x - 2}{4} - \dfrac{2 - x}{5}$Preview
  17. Q17$3x - 2 < 2x + 1$Preview
  18. Q18$5x - 3 > 3x - 5$Preview
  19. Q19$3(1 - x) < 2(x + 4)$Preview
  20. Q20$\dfrac{x}{2} \ge \dfrac{5x - 2}{3} - \dfrac{7x - 3}{5}$Preview
  21. Q21Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least…Preview
  22. Q22To receive Grade 'A' in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita's marks…Preview
  23. Q23Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.Preview
  24. Q24Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.Preview
  25. Q25The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of th…Preview
  26. Q26A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and…Preview

Miscellaneous Examples

+Miscellaneous Examplesi5 questions
  1. Example 9Solve $-8 \le 5x - 3 < 7$.Free
  2. Example 10Solve $-5 \le \dfrac{5 - 3x}{2} \le 8$.Free
  3. Example 11Solve the system of inequalities: $3x - 7 < 5 + x$ ... (1), $11 - 5x \le 1$ ... (2) and represent the solutions on the number line.Preview
  4. Example 12In an experiment, a solution of hydrochloric acid is to be kept between $30^\circ$ and $35^\circ$ Celsius. What is the range of temperature…Preview
  5. Example 13A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be added to it so that acid content in…Preview

Miscellaneous Exercise on Chapter 5

+Miscellaneous Exercisei14 questions
  1. Q1$2 \le 3x - 4 \le 5$Free
  2. Q2$6 \le -3(2x - 4) < 12$Free
  3. Q3$-3 \le 4 - \dfrac{7x}{2} \le 18$Free
  4. Q4$-15 < \dfrac{3(x - 2)}{5} \le 0$Preview
  5. Q5$-12 < 4 - \dfrac{3x}{-5} \le 2$Preview
  6. Q6$7 \le \dfrac{3x + 11}{2} \le 11$Preview
  7. Q7$5x + 1 > -24$, $5x - 1 < 24$Preview
  8. Q8$2(x - 1) < x + 5$, $3(x + 2) > 2 - x$Preview
  9. Q9$3x - 7 > 2(x - 6)$, $6 - x > 11 - 2x$Preview
  10. Q10$5(2x - 7) - 3(2x + 3) \le 0$, $2x + 19 \le 6x + 47$Preview
  11. Q11A solution is to be kept between $68^\circ$ F and $77^\circ$ F. What is the range in temperature in degree Celsius (C) if the Celsius / Fahr…Preview
  12. Q12A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less…Preview
  13. Q13How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more th…Preview
  14. Q14IQ of a person is given by the formula $IQ = \dfrac{MA}{CA} \times 100$, where MA is mental age and CA is chronological age. If $80 \le IQ \…Preview

Summary

- A linear inequality in one variable is of the form , , , or , where . The solution set is an interval on the number line.

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 41 questions41 questions
  1. Q1Solve for $x$: $\dfrac{4}{x+1} \le 3 \le \dfrac{6}{x+1}$, $(x > 0)$.Free
  2. Q2Solve for $x$: $\dfrac{|x-2|-1}{|x-2|-2} \le 0$.Free
  3. Q3Solve for $x$: $\dfrac{1}{|x|-3} \le \dfrac{1}{2}$.Free
  4. Q4Solve for $x$: $|x-1| \le 5$, $|x| \ge 2$.Preview
  5. Q5Solve for $x$: $-5 \le \dfrac{2-3x}{4} \le 9$.Preview
  6. Q6Solve for $x$: $4x + 3 \ge 2x + 17$, $3x - 5 < -2$.Preview
  7. Q7A company manufactures cassettes. Its cost and revenue functions are $C(x) = 26{,}000 + 30x$ and $R(x) = 43x$, respectively, where $x$ is th…Preview
  8. Q8The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between $8.2$ and $8.5$. If the…Preview
  9. Q9A solution of $9\%$ acid is to be diluted by adding $3\%$ acid solution to it. The resulting mixture is to be more than $5\%$ but less than…Preview
  10. Q10A solution is to be kept between $40^\circ\text{C}$ and $45^\circ\text{C}$. What is the range of temperature in degree fahrenheit, if the co…Preview
  11. Q11The longest side of a triangle is twice the shortest side and the third side is $2$ cm longer than the shortest side. If the perimeter of th…Preview
  12. Q12In drilling world's deepest hole it was found that the temperature $T$ in degree celcius, $x$ km below the earth's surface was given by $T =…Preview
  13. Q13Solve the following system of inequalities $\dfrac{2x+1}{7x-1} > 5$, $\dfrac{x+7}{x-8} > 2$.Preview
  14. Q14If $x < 5$, then (A) $-x < -5$ (B) $-x \le -5$ (C) $-x > -5$ (D) $-x \ge -5$Preview
  15. Q15Given that $x$, $y$ and $b$ are real numbers and $x < y$, $b < 0$, then (A) $\dfrac{x}{b} < \dfrac{y}{b}$ (B) $\dfrac{x}{b} \le \dfrac{y}{b}…Preview
  16. Q16If $-3x + 17 < -13$, then (A) $x \in (10, \infty)$ (B) $x \in [10, \infty)$ (C) $x \in (-\infty, 10]$ (D) $x \in [-10, 10)$Preview
  17. Q17If $x$ is a real number and $|x| < 3$, then (A) $x \ge 3$ (B) $-3 < x < 3$ (C) $x \le -3$ (D) $-3 \le x \le 3$Preview
  18. Q18$x$ and $b$ are real numbers. If $b > 0$ and $|x| > b$, then (A) $x \in (-b, \infty)$ (B) $x \in [-\infty, b)$ (C) $x \in (-b, b)$ (D) $x \i…Preview
  19. Q19If $|x - 1| > 5$, then (A) $x \in (-4, 6)$ (B) $x \in [-4, 6]$ (C) $x \in [-\infty, -4) \cup (6, \infty)$ (D) $x \in [-\infty, -4) \cup [6,…Preview
  20. Q20If $|x + 2| \le 9$, then (A) $x \in (-7, 11)$ (B) $x \in [-11, 7]$ (C) $x \in (-\infty, -7) \cup (11, \infty)$ (D) $x \in (-\infty, -7) \cup…Preview
  21. Q21On a number line, an open (unfilled) circle is drawn at $x=5$, and the line is shaded as a ray beginning just to the left of $5$ and extendi…Preview
  22. Q22On a number line, a filled (solid) circle is drawn at $x=\dfrac{9}{2}$, and the line is shaded as a ray beginning at $\dfrac{9}{2}$ and exte…Preview
  23. Q23On a number line, an open (unfilled) circle is drawn at $x=\dfrac{7}{2}$, and the line is shaded as a ray beginning just to the left of $\df…Preview
  24. Q24On a number line, a filled (solid) circle is drawn at $x=-2$, and the line is shaded as a ray beginning at $-2$ and extending indefinitely t…Preview
  25. Q25Fill in the blank: If $-4x \ge 12$, then $x$ ___ $-3$.Preview
  26. Q26Fill in the blank: If $\dfrac{-3}{4}x \le -3$, then $x$ ___ $4$.Preview
  27. Q27Fill in the blank: If $\dfrac{2}{x+2} > 0$, then $x$ ___ $-2$.Preview
  28. Q28Fill in the blank: If $x > -5$, then $4x$ ___ $-20$.Preview
  29. Q29Fill in the blank: If $x > y$ and $z < 0$, then $-xz$ ___ $-yz$.Preview
  30. Q30Fill in the blank: If $p > 0$ and $q < 0$, then $p - q$ ___ $p$.Preview
  31. Q31Fill in the blank: If $|x + 2| > 5$, then $x$ ___ $-7$ or $x$ ___ $3$.Preview
  32. Q32Fill in the blank: If $-2x + 1 \ge 9$, then $x$ ___ $-4$.Preview
  33. Q33State whether the following statement is True or False: If $x < y$ and $b < 0$, then $\dfrac{x}{b} < \dfrac{y}{b}$.Preview
  34. Q34State whether the following statement is True or False: If $xy > 0$, then $x > 0$ and $y < 0$.Preview
  35. Q35State whether the following statement is True or False: If $xy > 0$, then $x < 0$ and $y < 0$.Preview
  36. Q36State whether the following statement is True or False: If $xy < 0$, then $x < 0$ and $y < 0$.Preview
  37. Q37State whether the following statement is True or False: If $x < -5$ and $x < -2$, then $x \in (-\infty, -5)$.Preview
  38. Q38State whether the following statement is True or False: If $x < -5$ and $x > 2$, then $x \in (-5, 2)$.Preview
  39. Q39State whether the following statement is True or False: If $x > -2$ and $x < 9$, then $x \in (-2, 9)$.Preview
  40. Q40State whether the following statement is True or False: If $|x| > 5$, then $x \in (-\infty, -5) \cup [5, \infty)$.Preview
  41. Q41State whether the following statement is True or False: If $|x| \le 4$, then $x \in [-4, 4]$.Preview