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 products of variables that can be positive or negative, we need to consider cases where the product could be maximized (both positive, both negative) or minimized (one positive, one negative).
3. Test extreme value combinations: Evaluate \(\mathrm{xy}\) at the boundary values to find the actual minimum and maximum products, keeping in mind that the ranges are exclusive (values cannot equal the endpoints).
4. Verify the complete range: Confirm that all values between our calculated minimum and maximum are achievable by checking intermediate cases.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in plain English. We have two variables, \(\mathrm{x}\) and \(\mathrm{y}\), each with their own ranges:

• \(\mathrm{x}\) can be any value between -4 and 7 (but not exactly -4 or 7)
• \(\mathrm{y}\) can be any value between -6 and 3 (but not exactly -6 or 3)

Our job is to find what happens when we multiply any possible \(\mathrm{x}\) value with any possible \(\mathrm{y}\) value. Think of it like this: if we tried every possible combination of \(\mathrm{x}\) and \(\mathrm{y}\) values, what would be the smallest product we could get, and what would be the largest?

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

When we multiply two numbers, the result depends on their signs. Let's think through this intuitively:

• If both numbers are positive, we get a positive product
• If both numbers are negative, we also get a positive product
• If one is positive and one is negative, we get a negative product

Since \(\mathrm{x}\) can range from negative to positive (-4 to 7) and \(\mathrm{y}\) can also range from negative to positive (-6 to 3), we need to consider all these scenarios. The extreme values (biggest and smallest products) will likely occur when we use values very close to the boundaries of our ranges.

Process Skill: CONSIDER ALL CASES - Recognizing that we need to examine different sign combinations to find all extremes

3. Test extreme value combinations

Let's examine what happens at the boundary values. Since our ranges are exclusive (we can't use the exact endpoints), we'll consider values very close to the boundaries:

For maximum positive products:

• Largest positive \(\mathrm{x}\) (approaches 7) × Largest positive \(\mathrm{y}\) (approaches 3): \(\mathrm{xy}\) approaches \(7 \times 3 = 21\)
• Most negative \(\mathrm{x}\) (approaches -4) × Most negative \(\mathrm{y}\) (approaches -6): \(\mathrm{xy}\) approaches \((-4) \times (-6) = 24\)

For maximum negative products:

• Largest positive \(\mathrm{x}\) (approaches 7) × Most negative \(\mathrm{y}\) (approaches -6): \(\mathrm{xy}\) approaches \(7 \times (-6) = -42\)
• Most negative \(\mathrm{x}\) (approaches -4) × Largest positive \(\mathrm{y}\) (approaches 3): \(\mathrm{xy}\) approaches \((-4) \times 3 = -12\)

So our extreme values are:

• Most negative product: approaches -42
• Most positive product: approaches 24

Since we can't reach the exact boundary values, \(\mathrm{xy}\) can get arbitrarily close to -42 and 24 but never actually reach them.

4. Verify the complete range

We need to confirm that all values between -42 and 24 are actually achievable. This is true because:

• As \(\mathrm{x}\) and \(\mathrm{y}\) vary continuously within their ranges, \(\mathrm{xy}\) also varies continuously
• We can achieve \(\mathrm{xy} = 0\) by choosing \(\mathrm{x} = 0\) (which is in our range) and any \(\mathrm{y}\) value
• By choosing appropriate combinations of \(\mathrm{x}\) and \(\mathrm{y}\), we can achieve any value between our extremes

Therefore, \(\mathrm{xy}\) can take any value in the range \(-42 < \mathrm{xy} < 24\).

4. Final Answer

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

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

Answer: B

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the inequality constraints

Students often forget that the inequalities \(-4 < \mathrm{x} < 7\) and \(-6 < \mathrm{y} < 3\) use strict inequalities (< rather than ≤). This means \(\mathrm{x}\) cannot equal -4 or 7, and \(\mathrm{y}\) cannot equal -6 or 3. This leads them to incorrectly include these boundary values in their calculations, potentially affecting the final range.

2. Failing to consider all sign combinations systematically

Many students only test obvious combinations like the largest positive values or don't systematically consider what happens when one variable is positive and the other negative. They might miss testing scenarios like (negative \(\mathrm{x}\)) × (negative \(\mathrm{y}\)) or only focus on same-sign combinations, leading to an incomplete analysis of possible extreme values.

3. Assuming maximum occurs at intuitive combinations

Students often assume the maximum product occurs when both variables are at their largest positive values (\(\mathrm{x}\) approaching 7, \(\mathrm{y}\) approaching 3), missing that a larger product actually occurs when both variables are at their most negative values (\((-4) \times (-6) = 24\) vs \(7 \times 3 = 21\)).

Errors while executing the approach

1. Arithmetic errors in boundary calculations

When computing the extreme products, students frequently make sign errors or basic multiplication mistakes. For example, calculating \((-4) \times (-6)\) as -24 instead of +24, or \(7 \times (-6)\) as +42 instead of -42.

2. Incorrectly handling the strict inequality boundaries

Even when students recognize the inequalities are strict, they may incorrectly conclude that \(\mathrm{xy}\) can equal the boundary values (-42 or 24) rather than only approach them. This leads to using ≤ instead of < in their final answer.

Errors while selecting the answer

1. Confusing which extreme is the minimum and maximum

After calculating the boundary values correctly, students sometimes incorrectly order them, writing the range as \(-24 < \mathrm{xy} < 42\) instead of \(-42 < \mathrm{xy} < 24\), mixing up which calculated value represents the lower bound and which represents the upper bound.