Solve |`x|+|(4-x^2)/(x)|=|4/x|`

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: Rewrite the equation
We start with the given equation:
\[ |x| + \left| \frac{4 - x^2}{x} \right| = \left| \frac{4}{x} \right| \]

### Step 2: Simplify the absolute values
We can rewrite the left side:
\[ |x| + \left| \frac{4}{x} - x \right| = \left| \frac{4}{x} \right| \]
This gives us:
\[ |x| + \left| \frac{4}{x} - x \right| = \left| \frac{4}{x} \right| \]

### Step 3: Analyze the signs
For the absolute values to hold, we need to consider the signs of \( x \) and \( \frac{4 - x^2}{x} \). We can analyze the cases based on the value of \( x \).

### Step 4: Set up the inequality
We need to ensure that:
\[ x \cdot \left( \frac{4}{x} - x \right) \geq 0 \]
This simplifies to:
\[ 4 - x^2 \geq 0 \]
or
\[ x^2 \leq 4 \]

### Step 5: Solve the inequality
The inequality \( x^2 \leq 4 \) gives us:
\[ -2 \leq x \leq 2 \]

### Step 6: Exclude the point where \( x = 0 \)
Since \( x \) cannot be zero (it would make the original equation undefined), we exclude \( x = 0 \) from our solution. Thus, we have:
\[ x \in [-2, 0) \cup (0, 2] \]

### Final Solution
The solution to the equation is:
\[ x \in [-2, 0) \cup (0, 2] \]