Solution set of inequality `||x|-2| leq 3-|x|` consists of :

Options

A. a. exactly four integers
B. b. exactly five integers
C. c. two prime natural number
D. d. one prime natural number

Answer

To solve the inequality \( ||x| - 2| \leq 3 - |x| \), we will break it down step by step.

### Step 1: Analyze the inequality
We start with the inequality:
\[ ||x| - 2| \leq 3 - |x| \]

### Step 2: Break it down into cases
The expression \( ||x| - 2| \) can be analyzed based on the value of \( |x| \):
1. Case 1: \( |x| \geq 2 \)
2. Case 2: \( |x| < 2 \)

### Step 3: Solve Case 1: \( |x| \geq 2 \)
In this case, \( ||x| - 2| = |x| - 2 \). The inequality becomes:
\[ |x| - 2 \leq 3 - |x| \]
Adding \( |x| \) to both sides gives:
\[ 2|x| - 2 \leq 3 \]
Adding 2 to both sides:
\[ 2|x| \leq 5 \]
Dividing by 2:
\[ |x| \leq \frac{5}{2} \]
However, since we are in the case where \( |x| \geq 2 \), we combine these two results:
\[ 2 \leq |x| \leq \frac{5}{2} \]
This implies:
\[ - \frac{5}{2} \leq x \leq -2 \quad \text{or} \quad 2 \leq x \leq \frac{5}{2} \]

### Step 4: Solve Case 2: \( |x| < 2 \)
In this case, \( ||x| - 2| = 2 - |x| \). The inequality becomes:
\[ 2 - |x| \leq 3 - |x| \]
Adding \( |x| \) to both sides results in:
\[ 2 \leq 3 \]
This is always true, so for \( |x| < 2 \), the solution is:
\[ -2 < x < 2 \]

### Step 5: Combine the results
From Case 1, we have:
\[ - \frac{5}{2} \leq x \leq -2 \quad \text{or} \quad 2 \leq x \leq \frac{5}{2} \]
From Case 2, we have:
\[ -2 < x < 2 \]
Combining these results, we find the overall solution set:
\[ [-\frac{5}{2}, -2] \cup (-2, 2) \cup [2, \frac{5}{2}] \]

### Step 6: Identify the integer solutions
Now we need to find the integer solutions within the intervals:
- From \( [-\frac{5}{2}, -2] \): The integer is \( -2 \).
- From \( (-2, 2) \): The integers are \( -1, 0, 1 \).
- From \( [2, \frac{5}{2}] \): The integer is \( 2 \).

Thus, the integer solutions are:
\[ -2, -1, 0, 1, 2 \]

### Conclusion
The solution set consists of exactly 5 integers: \( -2, -1, 0, 1, 2 \).