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If \(-4 < \mathrm{x} < 7\) and \(-6 < \mathrm{y} < 3\), which of the following specifies all the possible values of \(\mathrm{xy}\)?
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:
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
When we multiply two numbers, the result depends on their signs. Let's think through this intuitively:
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
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:
For maximum negative products:
So our extreme values are:
Since we can't reach the exact boundary values, \(\mathrm{xy}\) can get arbitrarily close to -42 and 24 but never actually reach them.
We need to confirm that all values between -42 and 24 are actually achievable. This is true because:
Therefore, \(\mathrm{xy}\) can take any value in the range \(-42 < \mathrm{xy} < 24\).
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
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.
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.
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 \(7 \times 3 = 21\)).
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.
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.
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.