Write down the solution set for each inequality. 1 x - 1 2 > - 1 3 , x ∈ W

We have 1 x - 1 2 > - 1 3 Adding 1 2 to both sides, we get ⇒ 1 x - 1 2 + 1 2 > - 1 3 + 1 2 ⇒ 1 x > - 2 + 3 6 ⇒ 1 x > 1 6 If the reciprocal of each side of an inequality is taken, the symbol of the inequality is reversed. Thus, ⇒ x < 6 . . . i It is given that x = { 0 , 1 , 2 , 3 , 4 , 5 } . . . ( i i ) ∴ From ( i ) and ( i i ) , solution set is given by { 0 , 1 , 2 , 3 , 4 , 5 } . Hence, the required solution set is { 0 , 1 , 2 , 3 , 4 , 5 } .

Write down the solution set for each inequality. 3 x 4 - 1 3 < x 3 + 2 3 , where the replacement set is - 2 , - 1 , 0 , 1 , 2 , 3 , 5 , 7

We have 3 x 4 - 1 3 < x 3 + 2 3 Adding 1 3 to both sides, we get ⇒ 3 x 4 - 1 3 + 1 3 < x 3 + 2 3 + 1 3 ⇒ 3 x 4 < x 3 + 1 Subtracting x 3 from both sides, we get ⇒ 3 x 4 - x 3 < x 3 + 1 - x 3 ⇒ 9 x - 4 x 12 < 1 ⇒ 5 x 12 < 1 Multiplying by 12 5 on both sides, we get ⇒ 5 x 12 × 12 5 < 1 × 12 5 ⇒ x < 12 5 . . . ( i ) The replacement set is - 2 , - 1 , 0 , 1 , 2 , 3 , 5 , 7 Therefore, the solution set is - 2 , - 1 , 0 , 1 , 2 Hence, the required solution set is - 2 , - 1 , 0 , 1 , 2

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 .