The solution set of the inequation
`(3)/(|x|+2) ge 1`, is

Options

A. `[-1,1]`
B. `(-1, 1)`
C. `(-oo, 1]`
D. `[1, oo)`

Explanation

To solve the inequation \( \frac{3}{|x| + 2} \geq 1 \), we can follow these steps:

### Step 1: Rewrite the Inequation
Start with the given inequation:
\[ \frac{3}{|x| + 2} \geq 1 \]

### Step 2: Eliminate the Fraction
To eliminate the fraction, multiply both sides by \( |x| + 2 \). Since \( |x| + 2 \) is always positive (as the absolute value is non-negative and we are adding 2), the direction of the inequality remains the same:
\[ 3 \geq |x| + 2 \]

### Step 3: Rearrange the Inequation
Now, rearrange the inequality to isolate \( |x| \):
\[ 3 - 2 \geq |x| \]
\[ 1 \geq |x| \]
or
\[ |x| \leq 1 \]

### Step 4: Interpret the Absolute Value
The inequality \( |x| \leq 1 \) means that \( x \) is within the range of -1 to 1. Therefore, we can write:
\[ -1 \leq x \leq 1 \]

### Step 5: Write the Solution Set
The solution set can be expressed in interval notation as:
\[ [-1, 1] \]

### Final Answer
The solution set of the inequation \( \frac{3}{|x| + 2} \geq 1 \) is:
\[ [-1, 1] \]

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