If -{4 and (-6 , which of the following specifies all the possible values of (xy}?

1. Translate the problem requirements: We need to find all possible values that the product \(\mathrm{xy}\) can take when \(\mathrm{x}\) is between -4 and 7 (exclusive) and \(\mathrm{y}\) is between -6 and 3 (exclusive). We're looking for the minimum and maximum possible values of \(\mathrm{xy}\).
2. Identify critical boundary scenarios: Since we're dealing with a product of two variables with given ranges, the extreme values of \(\mathrm{xy}\) will occur when we multiply values close to the boundaries of the given ranges.
3. Analyze sign combinations for extreme values: Consider when \(\mathrm{xy}\) is maximized (both positive or both negative with large absolute values) and when \(\mathrm{xy}\) is minimized (one positive, one negative with large absolute values).
4. Calculate boundary products: Multiply the boundary values to find the theoretical limits, then adjust for the open intervals to determine the actual range of \(\mathrm{xy}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in everyday terms. We have two variables, \(\mathrm{x}\) and \(\mathrm{y}\), and each one can take on a range of values. Think of it like this: \(\mathrm{x}\) can be any number between -4 and 7 (but not exactly -4 or 7), and \(\mathrm{y}\) can be any number between -6 and 3 (but not exactly -6 or 3).

Our job is to figure out: when we multiply any possible value of \(\mathrm{x}\) with any possible value of \(\mathrm{y}\), what's the smallest product we could get? What's the largest product we could get?

For example, if \(\mathrm{x}\) could be 2 and \(\mathrm{y}\) could be 1, then \(\mathrm{xy}\) would be 2. If \(\mathrm{x}\) could be -3 and \(\mathrm{y}\) could be 2, then \(\mathrm{xy}\) would be -6. We need to find the complete range of all such possible products.

Process Skill: TRANSLATE - Converting the inequality constraints into a clear understanding of what range of products we need to find.

2. Identify critical boundary scenarios

Here's a key insight: when you're multiplying two numbers, the extreme products (biggest and smallest) usually happen when you use the extreme values of your original numbers.

Think about it this way - if you want the biggest possible product, you'd want to multiply the biggest possible numbers together. If you want the smallest product, you'd multiply in a way that gives you the most negative result.

So let's identify our boundary values:
• \(\mathrm{x}\) can get very close to -4 (but not equal to -4)
• \(\mathrm{x}\) can get very close to 7 (but not equal to 7)
• \(\mathrm{y}\) can get very close to -6 (but not equal to -6)
• \(\mathrm{y}\) can get very close to 3 (but not equal to 3)

The extreme products will happen when we pair these boundary values in different combinations.

3. Analyze sign combinations for extreme values

Now let's think about when products are biggest and smallest. This depends on the signs of the numbers we're multiplying:

Case 1: Both numbers positive
When both \(\mathrm{x}\) and \(\mathrm{y}\) are positive, we get a positive product. The largest positive product happens when both numbers are as large as possible.
• \(\mathrm{x}\) approaches 7, \(\mathrm{y}\) approaches 3
• Product approaches \(7 \times 3 = 21\)

Case 2: Both numbers negative
When both \(\mathrm{x}\) and \(\mathrm{y}\) are negative, we also get a positive product.
• \(\mathrm{x}\) approaches -4, \(\mathrm{y}\) approaches -6
• Product approaches \((-4) \times (-6) = 24\)

Case 3: One positive, one negative
This gives us negative products. The most negative (smallest) products happen when we multiply the largest positive with the most negative, or vice versa.
• \(\mathrm{x}\) approaches 7, \(\mathrm{y}\) approaches -6: Product approaches \(7 \times (-6) = -42\)
• \(\mathrm{x}\) approaches -4, \(\mathrm{y}\) approaches 3: Product approaches \((-4) \times 3 = -12\)

Process Skill: CONSIDER ALL CASES - Systematically examining all sign combinations to ensure we don't miss any extreme values.

4. Calculate boundary products

Let's calculate all the boundary products we identified:

1. \((+7) \times (+3) = 21\)
2. \((-4) \times (-6) = 24\)
3. \((+7) \times (-6) = -42\)
4. \((-4) \times (+3) = -12\)

Looking at all these results:
• The largest product we can get is 24
• The smallest (most negative) product we can get is -42

However, there's an important detail: since our original inequalities use < (not ≤), \(\mathrm{x}\) can never actually equal -4 or 7, and \(\mathrm{y}\) can never actually equal -6 or 3. This means \(\mathrm{xy}\) can get arbitrarily close to these boundary values but can never actually reach them.

Therefore: \(-42 < \mathrm{xy} < 24\)

The range of all possible values of \(\mathrm{xy}\) is greater than -42 and less than 24.

5. Final Answer

Comparing our result \(-42 < \mathrm{xy} < 24\) with the answer choices:

1. \(-42 < \mathrm{xy} < 21\)
2. \(-42 < \mathrm{xy} < 24\)
3. \(-28 < \mathrm{xy} < 18\)
4. \(-24 < \mathrm{xy} < 21\)
5. \(-24 < \mathrm{xy} < 24\)

Our calculated range exactly matches choice B.

Answer: B

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the inequality constraints
Students often confuse the strict inequalities (< and >) with inclusive inequalities (≤ and ≥). They might think that \(\mathrm{x}\) can actually equal -4 or 7, and \(\mathrm{y}\) can actually equal -6 or 3, leading them to believe that \(\mathrm{xy}\) can reach the boundary values like 24 or -42 exactly. This fundamental misreading affects their entire approach to finding the range.

2. Assuming only adjacent boundary values matter
Some students think they only need to check combinations like (\(\mathrm{x}\) approaching 7 with \(\mathrm{y}\) approaching 3) or (\(\mathrm{x}\) approaching -4 with \(\mathrm{y}\) approaching -6), missing the crucial insight that they need to systematically check ALL possible combinations of boundary values to find both the maximum and minimum products.

3. Forgetting to consider negative × negative = positive
Students might focus only on obvious cases like positive × positive = positive, but overlook that when both \(\mathrm{x}\) and \(\mathrm{y}\) are negative (\(\mathrm{x}\) approaching -4, \(\mathrm{y}\) approaching -6), the product is actually positive and could potentially be the maximum value. This leads to incomplete analysis of all possible extreme cases.

Errors while executing the approach

1. Sign errors in multiplication
When calculating products like \((-4) \times (-6)\) or \(7 \times (-6)\), students frequently make basic sign errors, especially with the negative × negative = positive rule. For example, they might calculate \((-4) \times (-6)\) as -24 instead of +24, which would completely change which value represents the maximum.

2. Mixing up which product is larger
After calculating all four boundary products (21, 24, -42, -12), students sometimes incorrectly identify which one is the maximum. They might think 21 is larger than 24, or incorrectly rank the negative values, leading them to establish wrong bounds for their final range.

3. Incorrectly handling the strict inequality implications
Even when students correctly identify that the boundary values are 24 and -42, they might write the final answer as \(-42 \leq \mathrm{xy} \leq 24\) or \(-42 < \mathrm{xy} \leq 24\), forgetting that since the original constraints use strict inequalities, the products can only approach but never reach these extreme values.

Errors while selecting the answer

1. Choosing the wrong boundary values
Students might correctly calculate that they need something like 'negative number < \(\mathrm{xy}\) < positive number' but then select an answer choice with incorrect boundary values. For example, they might choose \(-42 < \mathrm{xy} < 21\) (choice A) instead of \(-42 < \mathrm{xy} < 24\) (choice B), possibly because 21 'feels' more familiar or they made an error in determining the maximum.

2. Selecting inclusive vs. exclusive inequalities inconsistently
Some students might arrive at the correct boundary values (-42 and 24) but then select an answer choice that uses different inequality symbols than what the problem requires, not carefully checking whether the boundaries should be included or excluded in the final answer.