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\| ≤ 3(ii) 2\|x\| = 5(iii) [x + [x + [x]]] = 9
Answer» (i) \|x\| ≤ 3 The solution set of \|x\| ≤ a is -a ≤ x ≤ a ∴ The required solution is -3 ≤ x ≤ 3 ∴ The solution set is [-3, 3] (ii) 2\|x\| = 5 ∴ \|x\| = 5/2 ∴ x = ± 5/2 (iii) [x + [x + [x]]] = 9 ∴ [x + [x] + [x] ] = 9 …….[[x + n] = [x] + n, if n is an integer] ∴ [x + 2[x]] = 9 ∴ [x] + 2[x] = 9 …..[[2[x] is an integer]] ∴ [x] = 3 ∴ x ∈ [3, 4)

Solve the following for x, where |x| is modulus function, [x] is the greatest integer function, {x} is a fractional part function.(i) |x| ≤ 3(ii) 2|x| = 5(iii) [x + [x + [x]]] = 9

Answer»

(i) |x| ≤ 3 The solution set of |x| ≤ a is -a ≤ x ≤ a

∴ The required solution is -3 ≤ x ≤ 3

∴ The solution set is [-3, 3]

(ii) 2|x| = 5

∴ |x| = 5/2

∴ x = ± 5/2

(iii) [x + [x + [x]]] = 9

∴ [x + [x] + [x] ] = 9 …….[[x + n] = [x] + n, if n is an integer]

∴ [x + 2[x]] = 9

∴ [x] + 2[x] = 9 …..[[2[x] is an integer]]

∴ [x] = 3

∴ x ∈ [3, 4)