Question

Given : P = {x : 5 < 2x – 1 ≤ 11, x ∈ R} and Q = {x : –1 ≤ 3 + 4x < 23, x ∈ I}.

Represent P and Q on the number line. Find P ∩ Q. 

[Hint: P = {x ∈ R : 3 < x ≤ 6} and Q = {x ∈ I : –1 ≤ x < 5} = {–1, 0, 1, 2, 3, 4} ∴ P ∩ Q = {4}.]

Answer 1

Given,

P = {x : 5 < 2x – 1 ≤ 11, x ∈ R}

⇒ 5 < 2x – 1 ≤ 11

Solving left side,

5 < 2x – 1

⇒ 6 < 2x

⇒ 3 < x

⇒ x > 3

Solving right side,

2x – 1 ≤ 11

⇒ 2x ≤ 12

⇒ x ≤ 6

∴ P = {x : x ∈ R, 3 < x ≤ 6}.

The graph of the solution set of P is represented by thick black line starting from 3 (not including 3) till 6 (including 6).

Given,

Q = {x : –1 ≤ 3 + 4x < 23, x ∈ I}

⇒ –1 ≤ 3 + 4x < 23

⇒ –1 ≤ 3 + 4x and 3 + 4x < 23

⇒ –4x ≤ 3 + 1 and 4x < 20

⇒ –4x ≤ 4 and x < 5

⇒ –x ≤ 1 and x < 5

⇒ x ≥ –1 and x < 5

∴ Q = {x : x ∈ I, –1 ≤ x < 5} = {–1, 0, 1, 2, 3, 4}.

The graph of the solution set of Q is represented by thick black dots.

∴ P ∩ Q = {4}