Hardest SAT Math Problems (updated for Digital SAT) - PrepMaven - Free Printable

To solve the given problem, we need to simplify or analyze the expression:

$$
x - 3 \sqrt[3]{\frac{3x - 11}{3x - 9}}
$$

Let's break this down step by step.

Step 1: Analyze the Expression

The expression is:
$$
x - 3 \sqrt[3]{\frac{3x - 11}{3x - 9}}
$$

Here, we have:
- A variable \( x \)
- A cube root term \( \sqrt[3]{\frac{3x - 11}{3x - 9}} \)
- A coefficient of 3 multiplying the cube root term

Step 2: Simplify the Fraction Inside the Cube Root

The fraction inside the cube root is:
$$
\frac{3x - 11}{3x - 9}
$$

We can factor out a common term from the denominator:
$$
3x - 9 = 3(x - 3)
$$

So the fraction becomes:
$$
\frac{3x - 11}{3(x - 3)}
$$

Step 3: Consider Possible Simplifications

At this point, the expression is:
$$
x - 3 \sqrt[3]{\frac{3x - 11}{3(x - 3)}}
$$

There isn't an obvious way to further simplify the fraction \( \frac{3x - 11}{3(x - 3)} \) algebraically without additional information or constraints on \( x \). However, we can explore specific values of \( x \) to see if the expression simplifies nicely.

Step 4: Check for Special Values of \( x \)

Let's test some values of \( x \) to see if the expression simplifies:

#### Case 1: \( x = 3 \)
If \( x = 3 \):
- The denominator \( 3x - 9 \) becomes \( 3(3) - 9 = 0 \), which makes the fraction undefined.
- Therefore, \( x = 3 \) is not a valid solution.

#### Case 2: \( x = 4 \)
If \( x = 4 \):
- The numerator \( 3x - 11 \) becomes \( 3(4) - 11 = 12 - 11 = 1 \).
- The denominator \( 3x - 9 \) becomes \( 3(4) - 9 = 12 - 9 = 3 \).
- The fraction inside the cube root is:
$$
\frac{3x - 11}{3x - 9} = \frac{1}{3}
$$
- The cube root of \( \frac{1}{3} \) is \( \sqrt[3]{\frac{1}{3}} \).
- The expression becomes:
$$
x - 3 \sqrt[3]{\frac{1}{3}} = 4 - 3 \sqrt[3]{\frac{1}{3}}
$$

This does not simplify further without numerical approximation.

Step 5: General Analysis

Without additional constraints or specific instructions, the expression remains in its current form:
$$
x - 3 \sqrt[3]{\frac{3x - 11}{3(x - 3)}}
$$

Final Answer

The expression cannot be simplified further without additional context or constraints. Therefore, the final answer is:

$$
\boxed{x - 3 \sqrt[3]{\frac{3x - 11}{3x - 9}}}
$$