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The expression given is:

\[
\frac{x + 3\sqrt{x}}{x - 7\sqrt{x}}
\]

We need to determine the set of all \(x\) for which this expression is defined, and provide the answer in interval notation.

### Step 1: Restrictions from the Square Root
The square root function \( \sqrt{x} \) is only defined for non-negative values of \( x \). Therefore, we require:
\[
x \geq 0
\]

### Step 2: Restrictions from the Denominator
The expression is undefined when the denominator is zero. So, we must solve for \( x \) such that:
\[
x - 7\sqrt{x} \neq 0
\]
Let's solve this equation:

1. Set \( x - 7\sqrt{x} = 0 \).
2. Rearrange to get \( x = 7\sqrt{x} \).
3. Square both sides to eliminate the square root:
\[
x^2 = 49x
\]
4. Rearrange this equation:
\[
x^2 - 49x = 0
\]
5. Factor the equation:
\[
x(x - 49) = 0
\]
6. So, \( x = 0 \) or \( x = 49 \).

Thus, the denominator is zero when \( x = 0 \) or \( x = 49 \), which means the expression is undefined for these values.

### Step 3: Final Answer in Interval Notation
Combining the restrictions:
- \( x \geq 0 \) (due to the square root),
- but \( x \neq 0 \) and \( x \neq 49 \) (due to the denominator),

The set of \( x \) for which the expression is defined is:
\[
(0, 49) \cup (49, \infty)
\]

Would you like further clarification on any part of this process?

### Relative Questions:
1. How do we handle expressions involving square roots in general?
2. What other conditions make a rational expression undefined?
3. How can factoring help in solving equations involving square roots?
4. What is interval notation, and how is it used?
5. What are the common restrictions when dealing with radical and rational functions?

### Tip:
Always check both the numerator and denominator when determining where a rational expression is defined, especially when square roots are involved!

Giving your answer in interval notation, give the set of all x for which the expression \(\frac{x + 3\sqrt{x}}{x - 7\sqrt{x}}\) is defined.

Explore how to determine the domain of the expression \(\frac{x + 3\sqrt{x}}{x - 7\sqrt{x}}\) with a detailed solution. Learn about the restrictions imposed by square roots and undefined denominators, including factoring techniques. Suitable for high school students.

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