Ncert Solutions for Class 11

Linear Inequalities

• Exercise 5.1

• Miscellaneous Exercise

Linear Inequalities

Exercise 5.1

1. Solve 24x < 100, when

(i) x is a natural number.

(ii) x is an integer.

2. Solve – 12x > 30, when

(i) x is a natural number.

(ii) x is an integer.

3. Solve 5x – 3 < 7, when

(i) x is an integer.

(ii) x is a real number.

4. Solve 3x + 8 > 2, when

(i) x is an integer.

(ii) x is a real number.

5. Solve for x: 4x + 3 < 5x + 7.

6. Solve for x: 3x – 7 > 5x – 1.

7. Solve for x: 3(x – 1) ≤ 2 (x – 3).

8. Solve for x: 3 (2 – x) ≥ 2 (1 – x).

9. Solve for x: \(x+\frac{x}{2}+\frac{x}{3}<11\).

10. Solve for x: \(\frac{x}{3}>\frac{x}{2}+1\).

11. Solve for x: \(\frac{3(x-2)}{5}\le\frac{5(2-x)}{3}\).

12. Solve for x: \(\frac{1}{2}(\frac{3x}{5}+4)\ge\frac{1}{3}(x-6)\).

13. Solve for x: 2 (2x + 3) – 10 < 6 (x – 2).

14. Solve for x: 37 – (3x + 5) ≥ 9x – 8 (x – 3).

15. Solve for x: \(\frac{x}{4}<\frac{(5x-2)}{3}-\frac{(7x-3)}{5}\).

16. Solve for x: \(\frac{(2x-1)}{3}\ge\frac{(3x-2)}{4}-\frac{(2-x)}{5}\).

17. Solve the inequality 3x – 2 < 2x + 1 and show the graph of the solution in each case on number line.

18. Solve the inequality 5x – 3 ≥ 3x – 5 and show the graph of the solution in each case on number line.

19. Solve the inequality 3 (1 – x) < 2 (x + 4) and show the graph of the solution in each case on number line.

20. Solve the inequality \(\frac{x}{2}\ge\frac{(5x-2)}{3}-\frac{(7x-3)}{5}\) and show the graph of the solution in each case on number line.

21. Ravi 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 60 marks.

22. To 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 in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

23. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

24. Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

25. The 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 the triangle is at least 61 cm, find the minimum length of the shortest side.

26. A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?

Linear Inequalities

Miscellaneous Exercise

1. Solve : \(2\le 3x-4\le 5\).

2. Solve : \(6≤-3(2x-4)<12\).

3. Solve : \(-3≤4-{7x \over 2}≤18\).

4. Solve : \(-15<{3(x-2) \over 5}≤0\).

5. Solve : \(-12<4-{3x \over -5}≤2\).

6. Solve : \(7≤{(3x+11)\over 2}≤11\).

7. Solve : \(5x+1>-24\) \(,5x-1<24 \) and represent the solution graphically on number line.

8. Solve : \(2(x-1){<}x+5,\) \(3(x + 2) > 2-x \) and represent the solution graphically on number line.

9. Solve : \(3x-7>2(x-6)\)\(,6-x>11-2x \) and represent the solution graphically on number line.

10. Solve : \(5(2x-7)-3(2x+3)≤0\) \(,2x+19≤6x+47\) and represent the solution graphically on number line.

11. A solution is to be kept between 68° F and 77° F. What is the range in temperature in degree Celsius (C) if the Celsius / Fahrenheit (F) conversion formula is given by \(F={9 \over 5}C+32 \) ?

12. A 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 than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

13. How 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 than 25% but less than 30% acid content?

14. IQ of a person is given by the formula \(IQ={MA \over CA}×100\), where MA is mental age and CA is chronological age. If \(80 ≤ IQ ≤ 140\) for a group of 12 years old children, find the range of their mental age.