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\| ≥ 5(ii) \|x – 4\| + \|x – 2\| = 3(iii) x2 + 7\|x\| + 12 = 0
Answer» (i) \|x + 4\| ≥ 5 The solution of \|x\| ≥ a is x ≤ -a or x ≥ a ∴ \|x + 4\| ≥ 5 gives ∴ x + 4 ≤ -5 or x + 4 ≥ 5 ∴ x ≤ -5 – 4 or x ≥ 5 – 4 ∴ x ≤ -9 or x ≥ 1 ∴ The solution set = (-∞, – 9] ∪ [1, ∞) (ii) \|x – 4\| + \|x – 2\| = 3 …..(i) Case I: x < 2 Equation (i) reduces to 4 – x + 2 – x = 3 …….[x < 2 < 4, x – 4 < 0, x – 2 < 0] ∴ 6 – 3 = 2x ∴ x = 3/2 Case II: 2 ≤ x < 4 Equation (i) reduces to 4 – x + x – 2 = 3 ∴ 2 = 3 (absurd) There is no solution in [2, 4) Case III: x ≥ 4 Equation (i) reduces to x – 4 + x – 2 = 3 ∴ 2x = 6 + 3 = 9 ∴ x = 9/2 ∴ x = 3/2, 9/2 are solutions. The solution set = {3/2, 9/2} (iii) x² + 7\|x\| + 12 = 0 ∴ (\|x\|)² + 7\|x\| + 12 = 0 ∴ (\|x\| + 3) (\|x\| + 4) = 0 ∴ There is no x that satisfies the equation. The solution set = { } or Φ

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| ≥ 5(ii) |x – 4| + |x – 2| = 3(iii) x2 + 7|x| + 12 = 0

Answer»

(i) |x + 4| ≥ 5

The solution of |x| ≥ a is x ≤ -a or x ≥ a

∴ |x + 4| ≥ 5 gives

∴ x + 4 ≤ -5 or x + 4 ≥ 5

∴ x ≤ -5 – 4 or x ≥ 5 – 4

∴ x ≤ -9 or x ≥ 1

∴ The solution set = (-∞, – 9] ∪ [1, ∞)

(ii) |x – 4| + |x – 2| = 3 …..(i)

Case I: x < 2 Equation (i) reduces to

4 – x + 2 – x = 3 …….[x < 2 < 4, x – 4 < 0, x – 2 < 0]

∴ 6 – 3 = 2x

∴ x = 3/2

Case II: 2 ≤ x < 4

Equation (i) reduces to

4 – x + x – 2 = 3

∴ 2 = 3 (absurd)

There is no solution in [2, 4)

Case III: x ≥ 4

Equation (i) reduces to

x – 4 + x – 2 = 3

∴ 2x = 6 + 3 = 9

∴ x = 9/2

∴ x = 3/2, 9/2 are solutions.

The solution set = {3/2, 9/2}

(iii) x² + 7|x| + 12 = 0

∴ (|x|)² + 7|x| + 12 = 0

∴ (|x| + 3) (|x| + 4) = 0

∴ There is no x that satisfies the equation.

The solution set = { } or Φ