P is the solution set of 7x - 2 > 4x + 1 and Q is the solution set of 9x - 45 ≥ 5(x - 5); where x ∈ R. Represent :

(i) P ∩ Q

(ii) P - Q

(iii) P ∩ Q' on different number lines.

Given,

P is the solution set of 7x - 2 > 4x + 1

Solving,

⇒ 7x - 2 > 4x + 1

⇒ 7x - 4x > 1 + 2

⇒ 3x > 3

Dividing both sides by 3 we get,

⇒ x > 1

Given,

Q is the solution set of 9x - 45 ≥ 5(x - 5)

Solving,

⇒ 9x - 45 ≥ 5(x - 5)

⇒ 9x - 45 ≥ 5x - 25

⇒ 9x - 5x ≥ -25 + 45

⇒ 4x ≥ 20

Dividing both sides by 4 we get,

⇒ x ≥ 5.

∴ P = {x : x > 1, x ∈ R} and Q = {x : x ≥ 5, x ∈ R}

(i) P ∩ Q = Numbers common between P and Q

∴ Solution set = {x : x ≥ 5, x ∈ R}

Solution on the number line is :

(ii) P - Q = Numbers which belong to P but do not belong to Q

∴ Solution set = {x : 1 < x < 5, x ∈ R}

Solution on the number line is :

(iii) P ∩ Q' = Numbers which belong to P but do not belong to Q

∴ Solution set = {x : 1 < x < 5, x ∈ R}

Solution on the number line is :