Find domain for `y=1/sqrt(abs(x)-x)`.

Answer

To find the domain of the function \( y = \frac{1}{\sqrt{|x| - x}} \), we need to ensure that the expression inside the square root is positive, as the square root of a negative number is not defined in the real number system, and the denominator cannot be zero.

### Step 1: Set up the inequality
We need to solve the inequality:
\[ |x| - x > 0 \]

### Step 2: Analyze the absolute value
The expression \( |x| \) can be defined in two cases based on the value of \( x \):

1. **Case 1:** When \( x \geq 0 \)
- Here, \( |x| = x \).
- The inequality becomes:
\[ x - x > 0 \implies 0 > 0 \]
- This is false, so there are no valid \( x \) values in this case.

2. **Case 2:** When \( x < 0 \)
- Here, \( |x| = -x \).
- The inequality becomes:
\[ -x - x > 0 \implies -2x > 0 \implies x < 0 \]
- This is true for all \( x < 0 \).

### Step 3: Conclusion on the domain
Since the inequality \( |x| - x > 0 \) holds true only when \( x < 0 \), we conclude that the domain of the function is:
\[ \text{Domain: } (-\infty, 0) \]

### Summary
The domain of the function \( y = \frac{1}{\sqrt{|x| - x}} \) is all negative real numbers, which can be expressed as:
\[ \text{Domain: } (-\infty, 0) \]