Question

Given: P {x : 5 < 2x – 1 ≤ 11, x∈R)
Q{x : – 1 ≤ 3 + 4x < 23, x∈I) where
R = (real numbers), I = (integers)
Represent P and Q on number line. Write down the elements of P ∩ Q. 

Answer 1

P = {x : 5 < 2x – 1 ≤ 11}
5 < 2x – 1 ≤ 11
5 < 2 x – 1 and 2x – 1 ≤ 11
– 2 x < – 5 – 1 and 2 x ≤ 11 + 1
– 2x < –  6 and 2x ≤ 12
–x < –3
x > 3 or 3 < x
∴ Solution set = 3 < x ≤ 6 - {4, 5, 6}
Solution set on number line.

Q = {–1 ≤ 3 + 4x < 23}
–1 ≤ 3 + 4 x < 23
–1 < 3 + 4x and 3 + 4 x < 23
–4x < 3 + 1 and 4x < 23 - 3
–4x < 4 and 4x < 20
–x < 1 and x < 5
x > – 1
–1 < x
∴ Solution set = {0, 1, 2, 3, 4}
∴ Solution set on number line