The number of integers satisfying the equation `|x|+|(4-x^(2))/(x)|=|(4)/(x)|` is

Options

A. `5`
B. `4`
C. `6`
D. `7`

Explanation

To solve the equation \( |x| + \left| \frac{4 - x^2}{x} \right| = \left| \frac{4}{x} \right| \), we will follow these steps:

### Step 1: Analyze the expression
We start with the equation:
\[ |x| + \left| \frac{4 - x^2}{x} \right| = \left| \frac{4}{x} \right| \]
We can rewrite the left-hand side:
\[ |x| + \left| \frac{4 - x^2}{x} \right| = |x| + \left| \frac{4}{x} - x \right| \]

### Step 2: Determine the conditions for the absolute values
For the absolute values to be valid, we need to consider the conditions under which the expressions inside the absolute values are non-negative or negative.

1. The term \( \frac{4 - x^2}{x} \) is defined for \( x \neq 0 \).
2. We need to ensure that \( 4 - x^2 \geq 0 \), which simplifies to:
\[ x^2 \leq 4 \implies -2 \leq x \leq 2 \]

### Step 3: Identify the intervals
From the above condition, we have:
\[ x \in [-2, 2] \quad \text{and} \quad x \neq 0 \]
Thus, the possible intervals for \( x \) are:
\[ [-2, -1], [-1, 0), (0, 1], [1, 2] \]

### Step 4: Solve for each interval
We will analyze the equation in the intervals \( [-2, -1] \), \( [-1, 0) \), \( (0, 1] \), and \( [1, 2] \).

1. **Interval \( [-2, -1] \)**:
- Here, \( |x| = -x \) and \( |4 - x^2| = 4 - x^2 \).
- The equation becomes:
\[ -x + \frac{4 - x^2}{-x} = \frac{4}{-x} \]
- Simplifying gives:
\[ -x - \frac{4 - x^2}{x} = -\frac{4}{x} \]
- This leads to:
\[ -x - \frac{4}{x} + x = 0 \implies 0 = 0 \]
- All values in this interval satisfy the equation.

2. **Interval \( [-1, 0) \)**:
- Here, \( |x| = -x \) and \( |4 - x^2| = 4 - x^2 \).
- The equation becomes:
\[ -x + \frac{4 - x^2}{-x} = \frac{4}{-x} \]
- This simplifies similarly to:
\[ 0 = 0 \]
- All values in this interval satisfy the equation.

3. **Interval \( (0, 1] \)**:
- Here, \( |x| = x \) and \( |4 - x^2| = 4 - x^2 \).
- The equation becomes:
\[ x + \frac{4 - x^2}{x} = \frac{4}{x} \]
- This simplifies to:
\[ x + \frac{4}{x} - \frac{x^2}{x} = \frac{4}{x} \implies x = 0 \]
- No values in this interval satisfy the equation.

4. **Interval \( [1, 2] \)**:
- Here, \( |x| = x \) and \( |4 - x^2| = 4 - x^2 \).
- The equation becomes:
\[ x + \frac{4 - x^2}{x} = \frac{4}{x} \]
- This simplifies similarly to:
\[ x + \frac{4}{x} - \frac{x^2}{x} = \frac{4}{x} \implies x = 0 \]
- No values in this interval satisfy the equation.

### Step 5: Count the integers
The integers satisfying the equation are:
- From \( [-2, -1] \): -2, -1
- From \( [-1, 0) \): None
- From \( (0, 1] \): None
- From \( [1, 2] \): 1, 2

Thus, the integers satisfying the equation are:
- -2, -1, 1, 2

### Final Answer
The total number of integers satisfying the equation is **4**.