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.

Given,

P = {x : 5 < 2x - 1 ≤ 11, x ∈ R}

Solving L.H.S. of the inequation,

⇒ 5 < 2x - 1

⇒ 2x - 1 > 5

⇒ 2x > 5 + 1

⇒ 2x > 6

⇒ x > $\dfrac{6}{2}$

⇒ x > 3 ……..(1)

Solving R.H.S. of the inequation,

⇒ 2x - 1 ≤ 11

⇒ 2x ≤ 11 + 1

⇒ 2x ≤ 12

⇒ x ≤ $\dfrac{12}{2}$

⇒ x ≤ 6 ………(2)

From (1) and (2) we get,

3 < x ≤ 6

Since, x ∈ R

P = {x : 3 < x ≤ 6, x ∈ R}

Given,

Q = {x : -1 ≤ 3 + 4x < 23, x ∈ I}.

Solving L.H.S. of the inequation,

⇒ -1 ≤ 3 + 4x

⇒ 3 + 4x ≥ -1

⇒ 4x ≥ -1 - 3

⇒ 4x ≥ -4

⇒ x ≥ $-\dfrac{4}{4}$

⇒ x ≥ -1 ………(3)

Solving R.H.S. of the inequation,

⇒ 3 + 4x < 23

⇒ 4x < 23 - 3

⇒ 4x < 20

⇒ x < $\dfrac{20}{4}$

⇒ x < 5 ………..(4)

From (3) and (4) we get,

-1 ≤ x < 5

Since, x ∈ I

Q = {-1, 0, 1, 2, 3, 4}

P ∩ Q = Numbers common between P and Q = {4}

Hence, P ∩ Q = {4}.