The set of all real x satisfying the inequality `(3-|x|)/(4-|x|)ge 0`

Options

A. `[-3, 3] cup (-oo, -4) cup (4, oo)`
B. `(-oo, -4) cup (4, oo)`
C. `(-oo, -3) cup (4, oo)`
D. `(-oo, -3) cup (3, oo)`

Answer

To solve the inequality \(\frac{3 - |x|}{4 - |x|} \geq 0\), we will break it down into manageable steps.

### Step-by-Step Solution:

**Step 1: Identify when the expression is non-negative.**

The expression \(\frac{3 - |x|}{4 - |x|}\) is non-negative when both the numerator and denominator are either both positive or both negative.

**Step 2: Analyze the numerator and denominator separately.**

1. **Numerator:** \(3 - |x| \geq 0\)
- This simplifies to \(|x| \leq 3\).

2. **Denominator:** \(4 - |x| > 0\)
- This simplifies to \(|x| < 4\).

**Step 3: Solve the inequalities.**

- From \(|x| \leq 3\), we can deduce:
\[ -3 \leq x \leq 3 \]

- From \(|x| < 4\), we can deduce:
\[ -4 < x < 4 \]

**Step 4: Find the intersection of the solutions.**

The solution set for the numerator is \([-3, 3]\) and for the denominator is \((-4, 4)\). The intersection of these two sets is:
\[ [-3, 3] \]

**Step 5: Consider the case when both the numerator and denominator are negative.**

1. **Numerator:** \(3 - |x| < 0\)
- This simplifies to \(|x| > 3\).

2. **Denominator:** \(4 - |x| < 0\)
- This simplifies to \(|x| > 4\).

**Step 6: Solve these inequalities.**

- From \(|x| > 3\), we can deduce:
\[ x < -3 \quad \text{or} \quad x > 3 \]

- From \(|x| > 4\), we can deduce:
\[ x < -4 \quad \text{or} \quad x > 4 \]

**Step 7: Find the intersection of these solutions.**

The solution set for \(|x| > 3\) is \((-\infty, -3) \cup (3, \infty)\) and for \(|x| > 4\) is \((-\infty, -4) \cup (4, \infty)\). The intersection of these two sets is:
\[ (-\infty, -4) \cup (4, \infty) \]

**Step 8: Combine the solutions.**

The final solution set combines the results from both cases:
\[ [-3, 3] \cup (-\infty, -4) \cup (4, \infty) \]

### Final Result:

Thus, the set of all real \(x\) satisfying the inequality \(\frac{3 - |x|}{4 - |x|} \geq 0\) is:
\[ (-\infty, -4) \cup [-3, 3] \cup (4, \infty) \]