Solve the following for x, where |x| is modulus function, [x] is the g

1.	Solve the following for x, where \|x\| is modulus function, [x] is the greatest integer function, {x} is a fractional part function.(i) {x} > 4(ii) {x} = o(iii) {x} = 0.5(iv) 2{x} = x + [x]
Answer» (i) {x} > 4 This is a meaningless statement as 0 ≤ {x} < 1 ∴ The solution set = { } or Φ (ii) {x} = 0 ∴ x is an integer ∴ The solution set is Z. (iii) {x} = 0.5 ∴ x = ….., -2.5, -1.5, -0.5, 0.5, 1.5, ….. ∴ The solution set = {x : x = n + 0.5, n ∈ Z} (iv) 2{x} = x + [x] = [x] + {x} + [x] ……[x = [x] + {x}] ∴ {x} = 2[x] R.H.S. is an integer ∴ L.H.S. is an integer ∴ {x} = 0 ∴ [x] = 0 ∴ x = 0

Solve the following for x, where |x| is modulus function, [x] is the greatest integer function, {x} is a fractional part function.(i) {x} > 4(ii) {x} = o(iii) {x} = 0.5(iv) 2{x} = x + [x]

Answer»

(i) {x} > 4

This is a meaningless statement as 0 ≤ {x} < 1

∴ The solution set = { } or Φ

(ii) {x} = 0

∴ x is an integer

∴ The solution set is Z.

(iii) {x} = 0.5

∴ x = ….., -2.5, -1.5, -0.5, 0.5, 1.5, …..

∴ The solution set = {x : x = n + 0.5, n ∈ Z}

(iv) 2{x} = x + [x]

= [x] + {x} + [x] ……[x = [x] + {x}]

∴ {x} = 2[x]

R.H.S. is an integer

∴ L.H.S. is an integer

∴ {x} = 0

∴ [x] = 0

∴ x = 0