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). This means finding 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 constraints, the extreme values will occur when we choose values of \(\mathrm{x}\) and \(\mathrm{y}\) that are as close as possible to the boundary values of their respective ranges.
3. Test scenarios for maximum product: Consider combinations that would give the largest positive product - this happens when both variables have the same sign and are as large as possible in absolute value.
4. Test scenarios for minimum product: Consider combinations that would give the most negative product - this happens when variables have opposite signs and are as large as possible in absolute value.
5. Establish the complete range: Verify that all values between the minimum and maximum are achievable and express the final range.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in plain terms. We have two variables, \(\mathrm{x}\) and \(\mathrm{y}\), and each has a range of possible values. Think of \(\mathrm{x}\) as any number between -4 and 7 (but not including -4 or 7 exactly), and \(\mathrm{y}\) as any number between -6 and 3 (but not including -6 or 3 exactly). We want to find what happens when we multiply any possible \(\mathrm{x}\) value by any possible \(\mathrm{y}\) value - specifically, what's the smallest and largest result we could get?

In mathematical terms: we need to find the range of \(\mathrm{xy}\) where \(-4 < \mathrm{x} < 7\) and \(-6 < \mathrm{y} < 3\).

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical understanding of what we're seeking

2. Identify critical boundary scenarios

Here's the key insight: when we're looking for extreme values of a product, the most extreme results happen when we use values as close as possible to the boundaries of our allowed ranges.

Think about it this way - if you want the biggest possible result from multiplying two numbers, you'd want to use the biggest numbers you can. If you want the most negative result, you'd want one large positive and one large negative number.

So we need to consider what happens when \(\mathrm{x}\) gets close to -4 or close to 7, and when \(\mathrm{y}\) gets close to -6 or close to 3. Since our ranges are open (don't include the endpoints), we can get arbitrarily close to these boundary values.

3. Test scenarios for maximum product

To get the largest positive product, we want both numbers to have the same sign and be as large as possible in absolute value.

Let's consider the possibilities:
- Both positive and large: \(\mathrm{x}\) approaching 7, \(\mathrm{y}\) approaching 3
- Both negative and large in absolute value: \(\mathrm{x}\) approaching -4, \(\mathrm{y}\) approaching -6

For the first case: As \(\mathrm{x}\) approaches 7 and \(\mathrm{y}\) approaches 3, \(\mathrm{xy}\) approaches \(7 \times 3 = 21\)
For the second case: As \(\mathrm{x}\) approaches -4 and \(\mathrm{y}\) approaches -6, \(\mathrm{xy}\) approaches \((-4) \times (-6) = 24\)

Since \(24 > 21\), the maximum value that \(\mathrm{xy}\) can approach is 24.

4. Test scenarios for minimum product

To get the most negative product, we want the numbers to have opposite signs and be as large as possible in absolute value.

Let's consider the possibilities:
- Large positive \(\mathrm{x}\), large negative \(\mathrm{y}\): \(\mathrm{x}\) approaching 7, \(\mathrm{y}\) approaching -6
- Large negative \(\mathrm{x}\), large positive \(\mathrm{y}\): \(\mathrm{x}\) approaching -4, \(\mathrm{y}\) approaching 3

For the first case: As \(\mathrm{x}\) approaches 7 and \(\mathrm{y}\) approaches -6, \(\mathrm{xy}\) approaches \(7 \times (-6) = -42\)
For the second case: As \(\mathrm{x}\) approaches -4 and \(\mathrm{y}\) approaches 3, \(\mathrm{xy}\) approaches \((-4) \times 3 = -12\)

Since \(-42 < -12\), the minimum value that \(\mathrm{xy}\) can approach is -42.

5. Establish the complete range

Now we need to verify that all values between our minimum and maximum are actually achievable, and express our final range correctly.

Since we can choose any \(\mathrm{x}\) in \((-4, 7)\) and any \(\mathrm{y}\) in \((-6, 3)\), and the product function is continuous, every value between our minimum approach value of -42 and our maximum approach value of 24 can indeed be achieved.

However, since our original ranges are open intervals (don't include endpoints), the products -42 and 24 can be approached but never actually reached.

Therefore, the range of \(\mathrm{xy}\) is: \(-42 < \mathrm{xy} < 24\)

Process Skill: APPLY CONSTRAINTS - Ensuring we respect the open interval nature of the original constraints

4. Final Answer

The range of possible values for \(\mathrm{xy}\) is \(-42 < \mathrm{xy} < 24\).

Looking at our answer choices, this matches option B) \(-42 < \mathrm{xy} < 24\).

We can verify this makes sense: the extreme negative value comes from multiplying numbers close to 7 and -6, while the extreme positive value comes from multiplying numbers close to -4 and -6 (both negative, giving a positive product).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting open intervals as closed intervals: Students often overlook that \(-4 < \mathrm{x} < 7\) means \(\mathrm{x}\) cannot equal -4 or 7 exactly. They might incorrectly include the endpoints and calculate products like \((-4) \times (-6) = 24\) as an actual achievable value rather than an approached value. This leads to using ≤ instead of < in the final answer.

2. Focusing only on obvious extreme combinations: Students typically test only the most obvious combinations like largest positive × largest positive \((7 \times 3 = 21)\) without systematically considering all four corner scenarios. They miss that negative × negative can produce larger positive values, specifically that \((-4) \times (-6) = 24\) is larger than \(7 \times 3 = 21\).

3. Assuming symmetry in the product range: Students might incorrectly assume the range will be symmetric around zero or that the maximum positive and minimum negative values have the same absolute value. This misconception can lead them to expect ranges like \(-21 < \mathrm{xy} < 21\) rather than the correct asymmetric range.

Errors while executing the approach

1. Arithmetic errors in sign calculations: Students frequently make mistakes when multiplying negative numbers, particularly confusing \((-4) \times (-6) = 24\) with \((-4) \times 6 = -24\), or calculating \(7 \times (-6) = 42\) instead of -42. These sign errors directly affect which extreme values they identify.

2. Incomplete testing of boundary scenarios: Even when students understand they need to test extreme values, they often test only 2 or 3 of the 4 critical combinations. For example, they might test (7, 3) and (7, -6) combinations but forget to test (-4, 3) and (-4, -6) combinations, missing the true maximum or minimum values.

3. Incorrectly ordering the results: After calculating the four products (21, -42, 24, -12), students sometimes incorrectly identify which is the maximum or minimum. They might confuse -42 and -12 as to which is smaller, or fail to recognize that \(24 > 21\) for the maximum value.

Errors while selecting the answer

1. Confusing approached values with achievable values: Students correctly calculate that products approach -42 and 24, but then select answer choices with closed intervals like [-42, 24] or ranges that include the endpoints. They fail to maintain the strict inequality requirements from the original open intervals.

2. Mixing up maximum and minimum in final range: After determining that values approach -42 and 24, students sometimes write the range backwards as \(24 < \mathrm{xy} < -42\) or select answer choices where they've switched the bounds. This happens especially under time pressure when they lose track of which calculated value represents which extreme.