We need to determine for which values of $x$ the given mathematical expression is defined. The expression involves square roots, cube roots, and a fifth root, so we'll need to analyze each part of the expression to ensure that it is mathematically valid.

Expression:

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

1. First term: $2\sqrt{3x^2 - 7x + 4}$

The square root function is only defined for non-negative values. Therefore, we need to solve the inequality: $3x^2 - 7x + 4 \geq 0$ We will solve this quadratic inequality.

2. Second term: $\sqrt[3]{\frac{x - 2}{x + 2}}$

A cube root is defined for all real numbers, but we need to ensure that the denominator is not zero. Hence, we require: $x + 2 \neq 0 \quad \Rightarrow \quad x \neq -2$

3. Third term: $\frac{x}{\sqrt[5]{|x + 2|} - 3}$

For the expression inside the denominator, we need to ensure the fifth root is defined, and we also need to make sure the denominator does not equal zero: [ \sqrt[5]{|x + 2|} - 3 \neq 0 \quad \Rightarrow \quad \sqrt[5]{|x + 2|} \neq 3 ] This leads to: $|x + 2| \neq 243 \quad \Rightarrow \quad x \neq 241 \quad \text{and} \quad x \neq -245$

Summary of Conditions:

• $3x^2 - 7x + 4 \geq 0$
• $x \neq -2$
• $x \neq 241$
• $x \neq -245$

I will now solve the quadratic inequality and finalize the domain for $x$.The critical points of the quadratic equation $3x^2 - 7x + 4 = 0$ are $x = 1$ and $x = \frac{4}{3}$.

Now, we will test the intervals determined by these critical points to find where the inequality $3x^2 - 7x + 4 \geq 0$ holds true.The inequality $3x^2 - 7x + 4 \geq 0$ holds for the intervals:

• $(-\infty, 1]$
• $\left[\frac{4}{3}, \infty \right)$

Domain of the Expression

Considering all conditions:

1. $3x^2 - 7x + 4 \geq 0$ gives the domain $(-\infty, 1] \cup \left[\frac{4}{3}, \infty \right)$.
2. $x \neq -2$.
3. $x \neq 241$ and $x \neq -245$.

Thus, the final domain for which the entire expression is defined is: $(-\infty, -245) \cup (-245, -2) \cup (-2, 1] \cup \left[\frac{4}{3}, 241 \right) \cup (241, \infty)$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do we solve other types of root-based inequalities?
2. What is the difference between cube roots and square roots in terms of their domains?
3. How do we handle absolute values in solving inequalities?
4. Why does the denominator in rational expressions impose restrictions on the domain?
5. How does solving inequalities differ from solving equalities?

Tip: Always check for restrictions from denominators and radicals when determining the domain of a function.