Solve the following inequation and graph the solution set on the number line :

2x - 3 < x + 2 ≤ 3x + 5; x ∈ R.

Given,

2x - 3 < x + 2 ≤ 3x + 5

Solving L.H.S. of the equation,

⇒ 2x - 3 < x + 2

⇒ 2x - x < 2 + 3

⇒ x < 5 ………..(i)

Solving R.H.S. of the equation,

⇒ x + 2 ≤ 3x + 5

⇒ x - 3x ≤ 5 - 2

⇒ -2x ≤ 3

⇒ 2x ≥ -3

Dividing both sides by 2 we get,

⇒ x ≥ -1.5 ………(ii)

From (i) and (ii) we get,

-1.5 ≤ x < 5

∴ Solution set = {x : x ∈ R and -1.5 ≤ x < 5}.

Solution on the number line is :