Question

Find the complete set of values that satisfy the relations \[ |x - 3|<2 \quad \text{and} \quad |x| - 2|<3 \]

• A. \( (-5, 5) \)
• B. \( (-5, -1) \cup (1, 5) \)
• C. \( (1, 5) \)
• D. \( (-1, 1) \)

Hint:

Always solve compound inequalities one-by-one and then intersect their solution sets.

Answer 1

The Correct Option is B

Start with the first inequality: \[ |x - 3|<2 \Rightarrow -2<x - 3<2 \Rightarrow 1<x<5 \] Now the second inequality: \[ ||x| - 2|<3 \] Let \( y = |x| \), so: \[ |y - 2|<3 \Rightarrow -3<y - 2<3 \Rightarrow -1<y<5 \Rightarrow |x|<5 \] Also \( y = |x|>-1 \) is always true. So: \[ -5<x<5 \] Now intersect both conditions: From first: \( x \in (1, 5) \)
From second: \( x \in (-5, 5) \) excluding region where \( |x| - 2| \ge 3 \) → this is excluded when \( |x| \in (0,1) \cup (5,\infty) \)
So intersecting: \[ x \in (-5, -1) \cup (1, 5) \Rightarrow \boxed{(-5, -1) \cup (1, 5)} \]