If |x + 5| ≥ 10, then:

  1. x ∈ (15, 5]
  2. x ∈ (-5, 5]
  3. x ∈ (-∞, -15] ∪ [5, ∞)
  4. x ∈ (-∞, -5] ∪ [5, ∞)

Answer (Detailed Solution Below)

Option 3 : x ∈ (-∞, -15] ∪ [5, ∞)
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Detailed Solution

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Concept:

The Modulus Function '| |' is defined as:

 \(\rm |x|=\left\{\begin{matrix}\rm \ \ \ x, &\rm x \geq 0\\ \rm -x, &\rm x<0\\\end{matrix}\right.\)

Calculation:

We have two cases:

CASE I: If x + 5 ≥ 0, then |x + 5| = x + 5.

⇒ x + 5 ≥ 10

⇒ x ≥ 5

⇒ x ∈ [5, ∞)

CASE II: If x + 5 < 0, then |x + 5| = -(x + 5).

⇒ -(x + 5) ≥ 10

⇒ -x - 5 ≥ 10

⇒ -x ≥ 15

⇒ x ≤ -15

⇒ x ∈ (-∞, -15]

x ∈ (-∞, -15] ∪ [5, ∞).

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