# GSEB Solutions Class 11 Maths Chapter 6 Linear Inequalities Miscellaneous Exercise

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 6 Linear Inequalities Miscellaneous Exercise Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 6 Linear Inequalities Miscellaneous Exercise

Solve the inequalities in questions 1 to 6:
1. 2 â‰¤ 3x – 4 â‰¤ 5
2. 6 â‰¤ – 3(2x – 4) â‰¤ 12
3. – 3 â‰¤ 4 – $$\frac{7x}{2}$$ â‰¤ 18
4. – 15 â‰¤ $$\frac{3(x – 2)}{5}$$ â‰¤ 0
5. – 12 â‰¤ 4 – $$\frac{3x}{- 3}$$ â‰¤ 2
6. 7 â‰¤ $$\frac{3x+11}{2}$$ â‰¤ 11
Solutions to questions 1 – 6:
1. 2 â‰¤ 3x – 4 â‰¤ 5
2 + 4 â‰¤ 3x – 4 + 4 â‰¤ 5 + 4
or 6 â‰¤ 3x â‰¤ 9.
Dividing by 3, we get
2 â‰¤ x â‰¤ 3.
Thus, the solution is {2, 3}

2. 6 â‰¤ – 3(2x – 4) â‰¤ 12
or $$\frac{6}{3}$$ â‰¤ – (2x – 4) â‰¤ $$\frac{12}{3}$$ [Divide by 3]
or 2 â‰¤ – (2x – 4) â‰¤ 4
or 2 â‰¤ – 2x + 4 â‰¤ 4
or 2 – 4 â‰¤ – 2x + 4 – 4 â‰¤ 4 – 4
or – 2 â‰¤ – 2x â‰¤ – 0
or $$\frac{- 2}{- 2}$$ > $$\frac{- 2x}{- 2}$$ â‰¥ 0 [Divide by – 2]
or 1 â‰¥ x â‰¥ 0.
Hence, x is less than or equal to ‘1’ and is greater than 0, i.e; x âˆˆ [0, 1].

3. – 3 â‰¤ 4 – $$\frac{7x}{2}$$ â‰¤ 18
Subtracting 4, we get
– 3 – 4 â‰¤ 4 – $$\frac{7x}{2}$$ – 4 â‰¤ 18 – 4
or – 7 â‰¤ – $$\frac{7x}{2}$$ â‰¤ 14.
Dividing by – $$\frac{7}{2}$$, we get
– 7 Ã— (- $$\frac{2}{7}$$) â‰¥ $$\frac{- 7}{2}$$x Ã— (- $$\frac{2}{7}$$) â‰¥ 14 Ã— (- $$\frac{2}{7}$$).
2 â‰¥ x â‰¥ – 4 or – 4 â‰¤ x â‰¤ 2 or [- 4, 2]
âˆ´ x is less than or equal to 2 and greater than or equal to – 4.

4. – 15 < $$\frac{3(x – 2)}{5}$$ â‰¤ 0
– 15 Ã— 5 < $$\frac{3(x – 2)}{5}$$ Ã— 5 â‰¤ 0 Ã— 5 [Multiplying by 5]
or – 75 < 3(x – 2) â‰¤ 0
or – $$\frac{75}{3}$$ < x – 2 â‰¤ $$\frac{0}{3}$$ [Divide by 3]
or – 25 < x – 2 â‰¤ 0
or – 25 < x – 2 â‰¤ 0
or – 25 + 2 < x – 2 + 2 â‰¤ 0 + 2 [Add 2]
or – 23 < x â‰¤ 2.

5. – 12 < 4 – $$\frac{3x}{- 5}$$ â‰¤ 2.
Subtracting 4 from each side, we get
– 12 – 4 < 4 – $$\frac{3x}{- 5}$$ – 4 â‰¤ 2 – 4.
or – 16 < $$\frac{3x}{5}$$ â‰¤ – 2.
Multiplying each side by $$\frac{5}{3}$$, we get

6. 7 â‰¤ $$\frac{3x+11}{2}$$ â‰¤ 11.
Multiplying each side by 2, we get
7 Ã— 2 â‰¤ $$\frac{3x+11}{2}$$ Ã— 2 â‰¤ 11 Ã— 2.
or 14 â‰¤ 3x + 11 â‰¤ 22.
Subtracting 11, we get
14 – 11 â‰¤ 3x + 11 – 11 â‰¤ 22 – 11.
or 3 â‰¤ 3x â‰¤ 11.
Dividing by 3, we get
1 â‰¤ x â‰¤ $$\frac{11}{3}$$ or x âˆˆ [1, $$\frac{11}{3}$$].
i.e; x is greater than or equal to 1 and less than or equal to $$\frac{11}{3}$$.

Solve the inequalities from questions 7 to 10 and represent the solutions graphically on the number line:
7. 5x + 1 > – 24, 5x – 1 < 24
8. 2(x – 1) < x + 5, 3x + 2) > 2 – x
9. 3x – 7 > 2(x – 6), 6 – x > 11 – 2x
10. 5(2x – 7) – 3(2x + 3) â‰¤ 0, 2x + 19 â‰¤ 6x + 47
Solutions to questions from 7 to 11:
7. (i) 5x + 1 > – 24, subtracting 1, 5x > – 24 – 1
or 5x > – 25
âˆ´ Dividing by 5, x > – 5.

(ii) 5x – 1 < 24.
âˆ´ Adding 1, 5x < 24 + 1 = 25.
âˆ´ Dividing by 5, x < 5. x > – 5 and x < 5 are shown graphically as follows:

i.e., x is greater than – 5 and less than 5.

8. The inequalities are

Graph of the inequalities are shown below

i.e; x âˆˆ (- 1, 3) or x is greater than – 1 and less than 3.

9. The inequalities are 3x – 7 > 2(x – 6) and 6 – x > 11 – 2x.
or 3x – 7 > 2x – 12

Graphs of these inequalities are shown below:

x > 5 satisfies both the inequalities x > – 5 and x > 5
âˆ´ Solution is x > 5.

10. The inequalities are

Graphs of these inequalities are shown below:
x â‰¤ 11 i.e; x âˆˆ [- 7, 11]

Solution of x is greater than or equal to – 7 and x less or equal to 11.

11. A solution is to be kept between 68Â° F and 77Â°F. What is the range in temperature in degree Celsius (C), if the Celsius/Fahrenheit (F) conversion formula is given by
F = $$\frac{9}{5}$$C + 32
Solution:
We have:
F = $$\frac{9}{5}$$C + 32
But 68Â° < F < 77Â°.
âˆ´ 68Â° < $$\frac{9}{5}$$C + 32 < 77Â°.
Subtracting 32 from each side
68 – 32 < $$\frac{9}{5}$$C < 77 – 32
36 < $$\frac{9}{5}$$C < 45
Multiplying by $$\frac{5}{9}$$, we get
36 Ã— $$\frac{5}{9}$$ < C < 45 Ã— $$\frac{5}{9}$$
20 < C < 25. 12. A solution of 8% boric acid is to be diluted by adding 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 8% solution, how many litres of the 2% solution will have to be added? Solution: Let the 2% boric acid solution be x litres. âˆ´ Mixture is (640 + x) litres. or $$\frac{2x}{100}$$ + $$\frac{8 Ã— 640}{100}$$ > $$\frac{4}{100}$$ (640 + x)
or 2x + 5120 > 2560 + 4x
or 5120 – 2560 > 4x – 2x
or 2x < 2560.
or x < 1280
Also, 2% of x + 8% of 640 < 6% of (640 + x).
or $$\frac{2x}{100}$$ + $$\frac{8Ã—640}{100}$$ < $$\frac{6}{100}$$ (640 + x)
or 2x + 5120 < 3840 + 6x
or 5120 – 3840 < 6x – 2x or 4x > 1280 or x > 320.
Thus, the number of litres to be added should be greater than 320 litres.

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?
Solution:
Let x litres water is added to 45% solution of acid.
(i) âˆ´ 25% of (1125 + x) < $$\frac{1125Ã—45}{100}$$
or $$\frac{1125Ã—x}{4}$$ < $$\frac{1125Ã—9}{20}$$.
Multiplying by 20, we get
5(1125 + x) < 10125
or 5625 + 5x < 10125.
Subtracting 5625 from both sides, we get
âˆ´ 5x < 10125 – 5625 = 4500.
or x < $$\frac{4500}{5}$$ = 900 …………………. (1) (ii) Further 30% of (1125 + 4) > $$\frac{1125Ã—45}{100}$$.
or $$\frac{3}{10}$$ Ã— (1125 + x) > $$\frac{1125Ã—45}{100}$$ > $$\frac{1125Ã—9}{20}$$.
Multiplying both sides by 20, we get
6(1125 + x) > 1125 Ã— 9
or 6750 + 6x > 10125.
Subtracting 6750 from both sides, we get
6x > 10125 – 6750 > 3375
or x > $$\frac{3375}{6}$$ = 562.5 ……………….. (2)
From (1) and (2), we have:
562.5 < x < 900.

14. IQ of a person is given by the formula
IQ = $$\frac{MA}{CA}$$ Ã— 100,
where MA is mental age and CA chronological age. If 80 â‰¤ IQ â‰¤ 140 for a group of 12 years old children, find the range of their mental ages.
Solution:
We have:
IQ = $$\frac{MA}{CA}$$ Ã— 100
We are given 80 â‰¤ IQ â‰¤ 140.
Putting the values of IQ in it, we get
80 â‰¤ $$\frac{MA}{CA}$$ Ã— 100 â‰¤ 140.
Multiplying both sides by 12, we get
12 Ã— 80 â‰¤ MA Ã— 100 â‰¤ 140 Ã— 12
or 960 â‰¤ 100 Ã— MA â‰¤ 1680.
Dividing by 100, we get
9.6 â‰¤ MA â‰¤ 16.8.
Hence, mental age is greater than or equal to 9.6 and less than or equal to 16.8 years.