Which of the following CANNOT be a value of 1/(x-1)?

1. Translate the problem requirements: We need to determine which value from the answer choices is impossible for the expression \(\frac{1}{\mathrm{x}-1}\) to equal. This means finding which equation has no valid solution for x.
2. Test each answer choice by setting up equations: For each option, we'll set \(\frac{1}{\mathrm{x}-1}\) equal to that value and solve for x to see if a valid solution exists.
3. Identify restrictions on the original expression: Remember that x cannot equal 1 (since that would make the denominator zero), so any solution where \(\mathrm{x} = 1\) would be invalid.
4. Eliminate possibilities through algebraic solving: Work through each equation systematically, checking whether the resulting x-value creates a valid expression.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this question is really asking. We have an expression \(\frac{1}{\mathrm{x}-1}\), and we want to know which of the given values this expression can NEVER equal.

Think of it this way: imagine you're trying on different hats to see which ones fit. Here, we're trying different values to see which one our expression \(\frac{1}{\mathrm{x}-1}\) can never "wear" or equal.

The key insight is that if we can find a valid value of x that makes \(\frac{1}{\mathrm{x}-1}\) equal to one of our answer choices, then that choice IS possible. But if we can't find such an x (or if the x we find creates a problem), then that choice is impossible.

Process Skill: TRANSLATE - Converting the "cannot be a value" language into a systematic testing approach

2. Test each answer choice by setting up equations

Now let's test each answer choice by setting up the equation \(\frac{1}{\mathrm{x}-1} = \text{[each choice]}\) and solving for x.

Choice A: Can \(\frac{1}{\mathrm{x}-1} = -1\)?
Setting up: \(\frac{1}{\mathrm{x}-1} = -1\)
Multiply both sides by \((\mathrm{x}-1)\): \(1 = -1(\mathrm{x}-1)\)
Simplify: \(1 = -\mathrm{x} + 1\)
Solve: \(1 - 1 = -\mathrm{x}\), so \(0 = -\mathrm{x}\), therefore \(\mathrm{x} = 0\)
Check: When \(\mathrm{x} = 0\), we get \(\frac{1}{0-1} = \frac{1}{-1} = -1\) ✓

Choice C: Can \(\frac{1}{\mathrm{x}-1} = \frac{2}{3}\)?
Setting up: \(\frac{1}{\mathrm{x}-1} = \frac{2}{3}\)
Cross multiply: \(3 = 2(\mathrm{x}-1)\)
Expand: \(3 = 2\mathrm{x} - 2\)
Solve: \(3 + 2 = 2\mathrm{x}\), so \(5 = 2\mathrm{x}\), therefore \(\mathrm{x} = \frac{5}{2}\)
Check: When \(\mathrm{x} = \frac{5}{2}\), we get \(\frac{1}{\frac{5}{2}-1} = \frac{1}{\frac{3}{2}} = \frac{2}{3}\) ✓

Choice D: Can \(\frac{1}{\mathrm{x}-1} = 1\)?
Setting up: \(\frac{1}{\mathrm{x}-1} = 1\)
Multiply both sides by \((\mathrm{x}-1)\): \(1 = 1(\mathrm{x}-1)\)
Simplify: \(1 = \mathrm{x} - 1\)
Solve: \(1 + 1 = \mathrm{x}\), therefore \(\mathrm{x} = 2\)
Check: When \(\mathrm{x} = 2\), we get \(\frac{1}{2-1} = \frac{1}{1} = 1\) ✓

Choice E: Can \(\frac{1}{\mathrm{x}-1} = 2\)?
Setting up: \(\frac{1}{\mathrm{x}-1} = 2\)
Multiply both sides by \((\mathrm{x}-1)\): \(1 = 2(\mathrm{x}-1)\)
Expand: \(1 = 2\mathrm{x} - 2\)
Solve: \(1 + 2 = 2\mathrm{x}\), so \(3 = 2\mathrm{x}\), therefore \(\mathrm{x} = \frac{3}{2}\)
Check: When \(\mathrm{x} = \frac{3}{2}\), we get \(\frac{1}{\frac{3}{2}-1} = \frac{1}{\frac{1}{2}} = 2\) ✓

3. Identify restrictions on the original expression

Before we test Choice B, let's remember the fundamental restriction on our expression \(\frac{1}{\mathrm{x}-1}\).

In everyday terms: you can't divide by zero. It's like trying to share 1 cookie among 0 people - it doesn't make sense!

Mathematically: the expression \(\frac{1}{\mathrm{x}-1}\) is undefined when \(\mathrm{x}-1 = 0\), which means x cannot equal 1.

This restriction will be crucial as we examine the remaining choice.

4. Eliminate possibilities through algebraic solving

Now let's test the remaining choice:

Choice B: Can \(\frac{1}{\mathrm{x}-1} = 0\)?

Here's where our restriction becomes critical. Let's think about this logically first:

For a fraction to equal zero, the numerator must be zero while the denominator is non-zero. But in our expression \(\frac{1}{\mathrm{x}-1}\), the numerator is always 1, which is never zero.

Let's verify this algebraically:
Setting up: \(\frac{1}{\mathrm{x}-1} = 0\)
Multiply both sides by \((\mathrm{x}-1)\): \(1 = 0(\mathrm{x}-1)\)
Simplify: \(1 = 0\)

This gives us \(1 = 0\), which is impossible! This mathematical contradiction tells us that there is no value of x that can make \(\frac{1}{\mathrm{x}-1}\) equal to 0.

Process Skill: APPLY CONSTRAINTS - Recognizing that fractions with non-zero numerators can never equal zero

5. Final Answer

After testing all choices systematically:
• Choice A (-1): Possible when \(\mathrm{x} = 0\)
• Choice B (0): IMPOSSIBLE - creates the contradiction \(1 = 0\)
• Choice C (2/3): Possible when \(\mathrm{x} = \frac{5}{2}\)
• Choice D (1): Possible when \(\mathrm{x} = 2\)
• Choice E (2): Possible when \(\mathrm{x} = \frac{3}{2}\)

Therefore, the answer is B. The expression \(\frac{1}{\mathrm{x}-1}\) can never equal 0 because that would require the numerator 1 to equal 0, which is impossible.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "CANNOT be a value"
Students often confuse what the question is asking. Instead of finding which value is impossible for the expression \(\frac{1}{\mathrm{x}-1}\) to equal, they might look for which values ARE possible or try to find the domain restrictions directly. The key is understanding that we need to test each answer choice by setting up equations and seeing if they lead to valid solutions.

Faltering Point 2: Forgetting to consider domain restrictions
Students may jump straight into testing answer choices without first identifying that x cannot equal 1 (since this makes the denominator zero). This domain restriction is crucial for understanding why certain values might be impossible.

Faltering Point 3: Not recognizing this as a "reverse engineering" problem
Instead of working backwards from each answer choice (setting \(\frac{1}{\mathrm{x}-1}\) equal to each option), students might try to find the range of the function directly, which is much more complex and unnecessary for this problem.

Errors while executing the approach

Faltering Point 1: Algebraic manipulation errors
When solving equations like \(\frac{1}{\mathrm{x}-1} = -1\), students often make mistakes in cross-multiplication or sign handling. For example, they might forget the negative sign when expanding \(-1(\mathrm{x}-1) = -\mathrm{x} + 1\), or make errors when isolating x.

Faltering Point 2: Failing to verify solutions
After finding a value of x, students often don't substitute back into the original expression to check their work. This verification step is crucial to catch calculation errors and confirm the solution is correct.

Faltering Point 3: Misunderstanding the zero case
When testing if \(\frac{1}{\mathrm{x}-1}\) can equal 0, students might try to solve it algebraically without recognizing the logical impossibility. They may get confused by the contradiction \(1 = 0\) and not realize this definitively proves the impossibility.

Errors while selecting the answer

Faltering Point 1: Selecting a possible value instead of impossible
Since the question asks for what CANNOT be a value, students who find that most answer choices ARE possible might accidentally select one of those instead of the one that's impossible. This is especially likely if they're rushing or not carefully reading the question stem.

Faltering Point 2: Second-guessing the contradiction
When students get the mathematical contradiction \(1 = 0\) for choice B, they might think they made an error and try to "fix" it rather than recognizing that this contradiction is exactly what proves choice B is impossible.