Question

If \( 8 + 3x < |8 + 3x|, x \in \mathbb{R} \), then \( x \) lies in : 

• A. (-∞, ∞)
• B. \((-∞, \frac{-3}{8})\)
• C. \((-∞, \frac{-8}{3})\)
• D. \((-∞, \frac{8}{3})\)

Answer 1

The Correct Option is C

Given inequality: 
\( 8 + 3x < |8 + 3x| \)

Case 1: If \( 8 + 3x \geq 0 \), then \( |8 + 3x| = 8 + 3x \)
In this case, the inequality becomes:
\( 8 + 3x < 8 + 3x \), which is not true (equality, not strict inequality).

Case 2: If \( 8 + 3x < 0 \), then \( |8 + 3x| = -(8 + 3x) \)
So the inequality becomes:
\[ 8 + 3x < -(8 + 3x) \Rightarrow 8 + 3x < -8 - 3x \Rightarrow 6x < -16 \Rightarrow x < -\frac{8}{3} \]

Conclusion:
The inequality is satisfied when \( x < -\frac{8}{3} \)

Final Answer: \( x \in (-\infty, -\frac{8}{3}) \)