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The problem asks for the values of \( x \) where the given expression is defined. To find this, we need to analyze the conditions under which each part of the expression is defined. The expression is:

\[
2 \sqrt{3x^2 - 7x + 4} + \sqrt[3]{\frac{x - 2}{x + 2}} - \frac{x}{\sqrt[5]{|x + 2|} - 3}
\]

### Step-by-Step Analysis:
1. **Square Root \( \sqrt{3x^2 - 7x + 4} \):**
   The square root function is defined when the radicand (the expression inside the square root) is non-negative:
   \[
   3x^2 - 7x + 4 \geq 0
   \]

2. **Cube Root \( \sqrt[3]{\frac{x - 2}{x + 2}} \):**
   The cube root function is defined for all real numbers, but we need to ensure that the denominator \( x + 2 \neq 0 \), to avoid division by zero:
   \[
   x + 2 \neq 0 \quad \Rightarrow \quad x \neq -2
   \]

3. **Fifth Root \( \frac{x}{\sqrt[5]{|x + 2|} - 3} \):**
   The fifth root \( \sqrt[5]{|x + 2|} \) is defined for all real numbers because the absolute value ensures that \( |x + 2| \geq 0 \), but we must ensure that the denominator does not equal zero:
   \[
   \sqrt[5]{|x + 2|} - 3 \neq 0 \quad \Rightarrow \quad \sqrt[5]{|x + 2|} \neq 3
   \]

   Solving this:
   \[
   |x + 2| \neq 3^5 = 243
   \]
   This leads to two conditions:
   \[
   x + 2 \neq 243 \quad \Rightarrow \quad x \neq 241
   \]
   \[
   x + 2 \neq -243 \quad \Rightarrow \quad x \neq -245
   \]

### Putting All Conditions Together:
- The square root requires \( 3x^2 - 7x + 4 \geq 0 \).
- The cube root requires \( x \neq -2 \).
- The fifth root requires \( x \neq 241 \) and \( x \neq -245 \).

Now let's solve the quadratic inequality \( 3x^2 - 7x + 4 \geq 0 \). I'll compute this next.The quadratic inequality \( 3x^2 - 7x + 4 \geq 0 \) is solved as:

\[
x \in (-\infty, 1] \cup \left[\frac{4}{3}, \infty\right)
\]

### Combined Conditions:
1. \( x \in (-\infty, 1] \cup \left[\frac{4}{3}, \infty\right) \) from the square root condition.
2. \( x \neq -2 \) from the cube root condition.
3. \( x \neq 241 \) and \( x \neq -245 \) from the fifth root condition.

Thus, the final solution for the values of \( x \) where the expression is defined is:
\[
x \in (-\infty, 1] \cup \left[\frac{4}{3}, \infty\right) \quad \text{with} \quad x \neq -2, x \neq 241, x \neq -245
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve quadratic inequalities in general?
2. What are the steps to simplify expressions with cube roots?
3. How does the absolute value affect the domain of a function?
4. What is the significance of avoiding division by zero in expressions?
5. How do you analyze compound inequalities involving roots and absolute values?

### Tip:
When solving inequalities, always check for restrictions like division by zero, undefined roots, or invalid inputs for logarithmic or trigonometric functions.

При каких x определено выражение: 2√(3x² - 7x + 4) + ∛((x - 2) / (x + 2)) - x / (√⁵(|x + 2|) - 3)

Learn how to find the values of x that define the expression involving square roots, cube roots, and rational functions. This problem explores key algebraic concepts, including inequalities, radical functions, and absolute values. Suitable for students in Grades 10-12.

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