If x ∈ Z , find the solution set of the given equality. 2 3 x - 4 < 8

The given inequation is, 2 3 x - 4 < 8 . ⇒ 6 x - 8 < 8 Adding 8 to both the sides, we get, ⇒ 6 x - 8 + 8 < 8 + 8 ⇒ 6 x < 16 Dividing both the sides of the inequation by 6 , we get, ⇒ 6 x 6 < 16 6 ⇒ x < 2 . 666 Since x ∈ Z (set of integers). According to the given, We take integers, less than 2 . 666 . The solution set for the given inequation is, { . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 } . Hence, the solution set is { . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 } .

If x ∈ Z , find the solution set of the given equality. x 1 + 1 x > 5 x - 7

The given inequation is, x 1 + 1 x > 5 x - 7 . ⇒ x + 1 > 5 x - 7 Subtracting 1 from both the sides, we get, ⇒ x + 1 - 1 > 5 x - 7 - 1 ⇒ x > 5 x - 8 Subtracting 5 x from both the sides, we get, ⇒ x - 5 x > 5 x - 8 - 5 x ⇒ - 4 x > - 8 Dividing both the sides by - 4 , we get, ⇒ - 4 x - 4 < - 8 - 4 ⇒ x < 2 Since x ∈ Z (set of integers). According to the given, We take integers, less than 2 . The solution set for the given inequation is, { . . . , - 3 , - 2 , - 1 , 0 , 1 } . Hence, the solution set is { . . . , - 3 , - 2 , - 1 , 0 , 1 } .

Find the solution set of the given inequality. - 5 < x - 6 ≤ 3 - 2 x , x ∈ W

Given, - 5 < x - 6 ≤ 3 - 2 x , x ∈ W Consider, - 5 < x - 6 Adding 6 to both the sides, we get, ⇒ - 5 + 6 < x - 6 + 6 ⇒ 1 < x The solution set of - 5 < x - 6 is A = 2 , 3 , 4 , 5 , 6 , . . . Now consider, x - 6 ≤ 3 - 2 x Adding 6 to both the sides, we get, ⇒ x - 6 + 6 ≤ 3 - 2 x + 6 ⇒ x ≤ 9 - 2 x Adding 2 x to both the sides, we get, ⇒ x + 2 x ≤ 9 - 2 x + 2 x ⇒ 3 x ≤ 9 Dividing both the sides by 3 , we get, ⇒ 3 x 3 ≤ 9 3 ⇒ x ≤ 3 The solution set of x - 6 ≤ 3 - 2 x is B = 1 , 2 , 3 Thus, the solution set for - 5 < x - 6 ≤ 3 - 2 x is A ∩ B = 2 , 3 . Hence, the required solution set is 2 , 3 .

Find the solution set of the given inequality. - 7 < x - 6 ≤ 3 - 2 x , x ∈ Z

Given, - 7 < x - 6 ≤ 3 - 2 x , x ∈ Z Consider, - 7 < x - 6 Adding 6 to both the sides, we get, ⇒ - 7 + 6 < x - 6 + 6 ⇒ - 1 < x The solution set of - 7 < x - 6 is A = 0 , 1 , 2 , 3 , 4 , 5 , 6 , . . . Now consider, x - 6 ≤ 3 - 2 x Adding 6 to both the sides, we get, ⇒ x - 6 + 6 ≤ 3 - 2 x + 6 ⇒ x ≤ 9 - 2 x Adding 2 x to both the sides, we get, ⇒ x + 2 x ≤ 9 - 2 x + 2 x ⇒ 3 x ≤ 9 Dividing both the sides by 3 , we get, ⇒ 3 x 3 ≤ 9 3 ⇒ x ≤ 3 The solution set of x - 6 ≤ 3 - 2 x is B = . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 Thus, the solution set for - 7 < x - 6 ≤ 3 - 2 x is A ∩ B = 0 , 1 , 2 , 3 . Hence, the required solution set is 0 , 1 , 2 , 3 .

Find the solution set of the given inequality. - 21 < x - 20 ≤ - 2 x - 5 , x ∈ Z

Given, - 21 < x - 20 ≤ - 2 x - 5 , x ∈ Z Consider, - 21 < x - 20 Adding 20 to both the sides, we get, ⇒ - 21 + 20 < x - 20 + 20 ⇒ - 1 < x The solution set of ⇒ - 21 < x - 20 is A = 0 , 1 , 2 , 3 , 4 , . . . Now consider, x - 20 ≤ - 2 x - 5 Adding 20 to both the sides, we get, ⇒ x - 20 + 20 ≤ - 2 x - 5 + 20 ⇒ x ≤ - 2 x + 15 Adding 2 x to both the sides, we get, ⇒ x + 2 x ≤ - 2 x + 15 + 2 x ⇒ 3 x ≤ 15 Dividing both the sides by 3 , we get, ⇒ 3 x 3 ≤ 15 3 ⇒ x ≤ 5 The solution set of x - 20 ≤ - 2 x - 5 is B = - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , 4 , 5 Thus, the solution set for - 21 < x - 20 ≤ - 2 x - 5 is A ∩ B = 0 , 1 , 2 , 3 , 4 , 5 . Hence, the required solution set is 0 , 1 , 2 , 3 , 4 , 5 .

Find the solution set of the given inequality. 2 - x 3 < 4 x - 7 < 21 - 2 3 x , x ∈ N

Given, 2 - x 3 < 4 x - 7 < 21 - 2 3 x , x ∈ N Consider, 2 - x 3 < 4 x - 7 Adding 7 to both the sides, we get, ⇒ 2 - x 3 + 7 < 4 x - 7 + 7 ⇒ 9 - x 3 < 4 x Adding x 3 to both the sides, we get, ⇒ 9 - x 3 + x 3 < 4 x + x 3 ⇒ 9 < 13 x 3 Multiplying both the sides by 3 13 , we get, ⇒ 9 × 3 13 < 13 x 3 × 3 13 ⇒ 2 . 07 < x The solution set of 2 - x 3 < 4 x - 7 is A = 3 , 4 , 5 , 6 , . . . Now consider, 4 x - 7 < 21 - 2 3 x Adding 7 to both the sides, we get, ⇒ 4 x - 7 + 7 < 21 - 2 3 x + 7 ⇒ 4 x < 28 - 2 3 x Adding 2 x 3 to both the sides, we get, ⇒ 4 x + 2 3 x < 28 - 2 3 x + 2 3 x ⇒ 14 3 x < 28 Multiplying both the sides by 3 14 , we get, ⇒ 14 3 x × 3 14 < 28 × 3 14 ⇒ x < 6 The solution set of 4 x - 7 < 21 - 2 3 x is B = 1 , 2 , 3 , 4 , 5 Thus, the solution set for 2 - x 3 < 4 x - 7 < 21 - 2 3 x is A ∩ B = 3 , 4 , 5 . Hence, the required solution set is 3 , 4 , 5 .

Find the solution set of the given inequality. 3 - 7 x < - 6 x - 7 3 < 59 3 - 17 2 x , x ∈ N

Given, 3 - 7 x < - 6 x - 7 3 < 59 3 - 17 2 x , x ∈ N Consider, 3 - 7 x < - 6 x - 7 3 Adding 7 3 to both the sides, we get, ⇒ 3 - 7 x + 7 3 < - 6 x - 7 3 + 7 3 ⇒ - 7 x + 16 3 < - 6 x Adding 7 x to both the sides, we get, ⇒ - 7 x + 16 3 + 7 x < - 6 x + 7 x ⇒ 5 . 33 < x The solution set of 3 - 7 x < - 6 x - 7 3 is A = 6 , 7 , 8 , 9 , 10 , . . . Now consider, - 6 x - 7 3 < 59 3 - 17 2 x Adding 6 x to both the sides, we get, ⇒ - 6 x - 7 3 + 6 x < 59 3 - 17 2 x + 6 x ⇒ - 7 3 < 59 3 - 5 2 x Subtracting 59 3 from both the sides, we get, ⇒ - 7 3 - 59 3 < 59 3 - 5 2 x - 59 3 ⇒ - 22 < - 5 2 x Multiplying both the sides by - 2 5 , we get, ⇒ - 22 × - 2 5 > - 5 2 x × - 2 5 ⇒ 8 . 8 > x The solution set of - 6 x - 7 3 < 59 3 - 17 2 x is B = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 Thus, the solution set for 3 - 7 x < - 6 x - 7 3 < 59 3 - 17 2 x is A ∩ B = 6 , 7 , 8 . Hence, the required solution set is 6 , 7 , 8 .

E M B I B E

Be a Maths Star>Algebra>Chapter 10 - Inequalities>Exercise 10.1

Monica Capoor Solutions for Exercise 2: Exercise 10.1

Author:Monica Capoor

Monica Capoor Mathematics Solutions for Exercise - Monica Capoor Solutions for Exercise 2: Exercise 10.1

Attempt the free practice questions from Exercise 2: Exercise 10.1 with hints and solutions to strengthen your understanding. Be a Maths Star solutions are prepared by Experienced Embibe Experts.