1/(3 - 6/(x - 4)) The expression above is defined for which of the following values of x? x =...

1. Translate the problem requirements: Determine for which values of x the expression is "defined" - meaning it produces a real number (not undefined due to division by zero)
2. Identify potential undefined points: Look for all places where division by zero could occur in the complex fraction
3. Set up conditions for defined expression: Establish what values x cannot equal to avoid division by zero
4. Test each given value: Check whether x = 4, 6, and 8 satisfy the conditions for the expression to be defined

Execution of Strategic Approach

1. Translate the problem requirements

When we say an expression is "defined," we mean it gives us a real number as an answer. Think of it like a calculator - if you type in an expression and get an actual number (not an error message), then the expression is defined for those input values.

The key thing that makes expressions undefined is division by zero. Just like your calculator would show "ERROR" if you tried to divide by zero, mathematical expressions become undefined when any denominator equals zero.

So our job is to figure out: for which values of x (specifically x = 4, 6, and 8) does this complex fraction give us a real number answer?

Process Skill: TRANSLATE - Converting the concept of "defined" into the practical requirement of avoiding division by zero

2. Identify potential undefined points

Let's look at our expression: \(\frac{1}{3 - \frac{6}{\mathrm{x} - 4}}\)

This is a complex fraction - we have a fraction inside another fraction. We need to spot every place where division by zero could happen:

First potential problem: The inner fraction \(\frac{6}{\mathrm{x} - 4}\)
This becomes undefined when the denominator x - 4 = 0, which means x = 4.

Second potential problem: The outer fraction \(\frac{1}{3 - \frac{6}{\mathrm{x} - 4}}\)
This becomes undefined when the entire denominator equals zero: \(3 - \frac{6}{\mathrm{x} - 4} = 0\)

So we have two different ways this expression can break down, and we need to check both.

3. Set up conditions for defined expression

For our expression to be defined, we need to avoid both problems we identified:

Condition 1: x ≠ 4 (so the inner fraction doesn't have zero in denominator)

Condition 2: \(3 - \frac{6}{\mathrm{x} - 4} ≠ 0\) (so the outer fraction doesn't have zero in denominator)

Let's solve Condition 2. If \(3 - \frac{6}{\mathrm{x} - 4} = 0\), then:
\(3 = \frac{6}{\mathrm{x} - 4}\)

Cross-multiplying: \(3(\mathrm{x} - 4) = 6\)
\(3\mathrm{x} - 12 = 6\)
\(3\mathrm{x} = 18\)
\(\mathrm{x} = 6\)

So our expression is undefined when x = 4 OR when x = 6.
This means our expression IS defined when x ≠ 4 AND x ≠ 6.

4. Test each given value

Now let's check each option:

Testing x = 4:
When x = 4, we get \(\frac{6}{4-4} = \frac{6}{0}\), which is undefined.
So the entire expression is undefined when x = 4.
❌ Statement I is FALSE

Testing x = 6:
When x = 6: \(\frac{6}{6-4} = \frac{6}{2} = 3\)
So our expression becomes: \(\frac{1}{3-3} = \frac{1}{0}\), which is undefined.
❌ Statement II is FALSE

Testing x = 8:
When x = 8: \(\frac{6}{8-4} = \frac{6}{4} = 1.5\)
So our expression becomes: \(\frac{1}{3-1.5} = \frac{1}{1.5} = \frac{2}{3}\)
This gives us a real number, so the expression is defined.
✓ Statement III is TRUE

Process Skill: CONSIDER ALL CASES - Testing each given value systematically to avoid missing any scenarios

4. Final Answer

Only statement III is true - the expression is defined only for x = 8 among the given options.

The answer is C. III only.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "defined" means
Students often confuse "defined" with "solvable" or think it means finding when the expression equals zero. They may try to set the entire expression equal to zero instead of identifying when it produces a real number output. The key insight is that "defined" means avoiding division by zero anywhere in the expression.

2. Missing the nested fraction structure
Students may only focus on the obvious denominator (x - 4) and miss that there's a complex denominator in the outer fraction. They fail to recognize that \(3 - \frac{6}{\mathrm{x} - 4}\) is also a denominator that cannot equal zero, leading to incomplete analysis.

3. Confusing undefined points with defined points
Students may set up the problem backwards, looking for values where the expression equals zero rather than values where denominators equal zero. This leads them to find the wrong constraint conditions.

Errors while executing the approach

1. Algebraic errors when solving \(3 - \frac{6}{\mathrm{x} - 4} = 0\)
When cross-multiplying \(3 = \frac{6}{\mathrm{x} - 4}\) to get \(3(\mathrm{x} - 4) = 6\), students commonly make sign errors or arithmetic mistakes. For example, getting \(3\mathrm{x} - 12 = 6\) correct but then incorrectly calculating \(3\mathrm{x} = 6\) instead of \(3\mathrm{x} = 18\), leading to x = 2 instead of x = 6.

2. Incomplete checking of both undefined conditions
Students may correctly identify that x = 4 makes the inner fraction undefined but forget to check when the outer denominator equals zero. Or they may find one undefined point but not systematically verify both conditions for all test values.

Errors while selecting the answer

1. Selecting values that make the expression undefined
Students correctly identify that x = 4 and x = 6 make the expression undefined, but then mistakenly select answer choices that include these values, thinking these are the "correct" answers. They confuse what the question is asking for (defined values) with what they calculated (undefined values).