Solve 3x - 5 ≤ 6x + 4 < 11 + x, when

(i) x ∈ W

(ii) x ∈ Z

Represent the solution set on a real number in each case.

Given : 3x - 5 ≤ 6x + 4 < 11 + x

Solving L.H.S. of the inequation, we get :

⇒ 3x - 5 ≤ 6x + 4

⇒ 3x - 6x - 5 ≤ 4

⇒ -3x - 5 ≤ 4

⇒ -3x ≤ 4 + 5

⇒ -3x ≤ 9

⇒ 3x ≥ -9

⇒ x ≥ -$\dfrac{9}{3}$

⇒ x ≥ -3 …………………..(1)

Solving R.H.S. of the inequation, we get :

⇒ 6x + 4 < 11 + x

⇒ 6x + 4 - x < 11

⇒ 5x + 4 < 11

⇒ 5x < 11 - 4

⇒ 5x < 7

⇒ x < $\dfrac{7}{5}$ ………………….(2)

From (1) and (2), we get :

⇒ -3 ≤ x < $\dfrac{7}{5}$

⇒ -3 ≤ x < 1.4

(i) Since,

x ∈ W and -3 ≤ x < 1.4

Solution set = {0, 1}

Hence, solution set = {0, 1}.

(ii) Since,

x ∈ Z and -3 ≤ x < 1.4

Solution set = {-3, -2, -1, 0, 1}

Hence, solution set = {-3, -2, -1, 0, 1}.