The given expression is:

\[
\sqrt{-x} + 7
\]

For this expression to be valid in the real number system, the expression inside the square root (i.e., \(-x\)) must be non-negative. This is because square roots of negative numbers are not defined in the real numbers (but they are defined in the complex number system).

So, for \(\sqrt{-x}\) to be valid:
\[
-x \geq 0 \quad \text{or} \quad x \leq 0
\]

### Simplifying the Expression
1. **If \(x < 0\):**
   The square root of \(-x\) will be a real number, as \(-x\) becomes positive. For example, if \(x = -4\), then:
   \[
   \sqrt{-(-4)} + 7 = \sqrt{4} + 7 = 2 + 7 = 9
   \]
2. **If \(x = 0\):**
   The square root becomes \(\sqrt{-0} = 0\), so the expression simplifies to:
   \[
   0 + 7 = 7
   \]
   
3. **If \(x > 0\):** The square root becomes imaginary because \(-x\) is negative. In such cases, you can use imaginary numbers. For example, if \(x = 4\), the expression becomes:
   \[
   \sqrt{-4} + 7 = 2i + 7
   \]
   where \(i\) is the imaginary unit.

Would you like more details on how the square root works with imaginary numbers, or do you have any specific questions?

Here are five related questions for further practice:
1. What is the value of \(\sqrt{-9} + 7\)?
2. What is \(\sqrt{-(-5)} + 7\)?
3. Can you simplify \(\sqrt{-x}\) when \(x = 3i\)?
4. What happens if you try to evaluate \(\sqrt{-x} + 7\) for \(x > 0\)?
5. How do you simplify expressions involving square roots and imaginary numbers?

**Tip:** When dealing with square roots of negative numbers, remember that imaginary numbers are used, where \(i\) represents \(\sqrt{-1}\).

Learn how to simplify expressions like √(-x) + 7 by understanding square roots of negative numbers and when imaginary numbers apply. This problem covers the properties of radicals and complex numbers, suitable for students in Grades 9-12.

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