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 a step-by-step approach.

### Step 1: Identify the restrictions
First, we note that \( x \) cannot be equal to 0 because it would make the terms \( \frac{4 - x^2}{x} \) and \( \frac{4}{x} \) undefined. Therefore, we have:
\[ x \neq 0 \]

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ |x| + \left| \frac{4 - x^2}{x} \right| = \left| \frac{4}{x} \right| \]

### Step 3: Simplify the absolute values
We can express \( \left| \frac{4 - x^2}{x} \right| \) as:
\[ \left| \frac{4 - x^2}{x} \right| = \frac{|4 - x^2|}{|x|} \]
Thus, the equation becomes:
\[ |x| + \frac{|4 - x^2|}{|x|} = \frac{4}{|x|} \]

### Step 4: Multiply through by \( |x| \) (since \( x \neq 0 \))
Multiplying both sides by \( |x| \) gives:
\[ |x|^2 + |4 - x^2| = 4 \]

### Step 5: Analyze the cases for \( |4 - x^2| \)
We need to consider two cases based on the definition of absolute value:

**Case 1:** \( 4 - x^2 \geq 0 \) (i.e., \( x^2 \leq 4 \) or \( -2 \leq x \leq 2 \))
In this case, \( |4 - x^2| = 4 - x^2 \). The equation becomes:
\[ |x|^2 + 4 - x^2 = 4 \]
This simplifies to:
\[ |x|^2 - x^2 = 0 \]
Since \( |x|^2 = x^2 \) for all \( x \), this is always true. Thus, any \( x \) in the interval \( [-2, 2] \) satisfies the equation.

**Case 2:** \( 4 - x^2 < 0 \) (i.e., \( x^2 > 4 \) or \( x < -2 \) or \( x > 2 \))
In this case, \( |4 - x^2| = x^2 - 4 \). The equation becomes:
\[ |x|^2 + x^2 - 4 = 4 \]
This simplifies to:
\[ |x|^2 + x^2 = 8 \]
Since \( |x|^2 = x^2 \), we have:
\[ 2x^2 = 8 \]
Thus:
\[ x^2 = 4 \]
This gives us \( x = 2 \) or \( x = -2 \).

### Step 6: Combine the solutions
From Case 1, we have the interval \( [-2, 2] \) (excluding \( x = 0 \)). From Case 2, we found \( x = 2 \) and \( x = -2 \).

### Step 7: Identify the integer solutions
The integers satisfying the equation are:
- From the interval \( [-2, 2] \): \( -2, -1, 1, 2 \)

### Conclusion
Thus, the number of integers satisfying the equation is:
\[ \text{Total integers} = 4 \]