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If it is true that \(\mathrm{x} > -2\) and \(\mathrm{x} < 7\), which of the following must be true?
Let's start by understanding what we're given and what we need to find.
We're told that x satisfies two conditions at the same time:
Think of this like saying "x lives in the neighborhood between -2 and 7, but doesn't include the houses at exactly -2 or exactly 7."
Our job is to find which of the answer choices MUST always be true for every possible value that x could be. This means we're looking for a statement that can never be violated, no matter which specific value of x we pick from our allowed range.
Process Skill: TRANSLATE - Converting the problem language "which of the following must be true" into the mathematical understanding that we need a statement that's always true for every x in our range.
Now let's combine our two conditions to see exactly what values x can take.
Since \(\mathrm{x > -2}\) AND \(\mathrm{x < 7}\) (both must be true simultaneously), we can write this as:
\(\mathrm{-2 < x < 7}\)
Imagine this on a number line: x can be any number between -2 and 7, but not including -2 or 7 themselves. For example:
Process Skill: VISUALIZE - Using the number line to see the complete range of possible x values.
Now we'll check each answer choice to see if it MUST be true for every possible x in our range \(\mathrm{(-2, 7)}\).
Choice A: \(\mathrm{x > 2}\)
Is this always true? Let's test with \(\mathrm{x = 0}\) (which is in our allowed range).
\(\mathrm{0 > 2}\)? No, this is false.
Since we found a valid x value that makes this statement false, Choice A is not always true.
Choice B: \(\mathrm{x > -7}\)
Is this always true? Since our smallest possible x values are just slightly greater than -2, let's check:
If x is anything greater than -2, is x also greater than -7?
Yes! Since -2 is already much greater than -7, and \(\mathrm{x > -2}\), then x must definitely be greater than -7.
This looks promising - let's verify with examples:
Choice C: \(\mathrm{x < 2}\)
Is this always true? Let's test with \(\mathrm{x = 5}\) (which is in our allowed range).
\(\mathrm{5 < 2}\)? No, this is false.
Since we found a valid x value that makes this statement false, Choice C is not always true.
Choice D: \(\mathrm{-7 < x < 2}\)
This says x must be between -7 and 2. But we know x can be as large as 6.9 (just under 7).
\(\mathrm{6.9 < 2}\)? No, this is false.
Since we found valid x values that make this statement false, Choice D is not always true.
Process Skill: CONSIDER ALL CASES - Testing each choice with strategic examples from different parts of our allowed range to see if any choice can be violated.
From our testing:
Let's double-check Choice B logically: If \(\mathrm{x > -2}\), and \(\mathrm{-2 > -7}\), then by the transitive property, \(\mathrm{x > -7}\) must always be true.
Choice B (\(\mathrm{x > -7}\)) is the statement that MUST always be true.
This is because every value in our range \(\mathrm{(-2, 7)}\) is automatically greater than -7, since -7 is far to the left of -2 on the number line. No matter which specific value x takes within our allowed range, it will always satisfy \(\mathrm{x > -7}\).
The answer is B.
1. Misinterpreting "must be true" as "could be true"
Students often confuse these concepts. The question asks which statement MUST be true (meaning it's true for every possible x value), but students might look for statements that COULD be true (meaning it's true for at least one x value). This fundamental misunderstanding changes the entire approach to the problem.
2. Failing to properly combine the two given conditions
Students might treat \(\mathrm{x > -2}\) and \(\mathrm{x < 7}\) as separate, independent conditions rather than understanding they must both be satisfied simultaneously. This leads to analyzing each condition in isolation instead of recognizing that x must fall within the intersection: \(\mathrm{-2 < x < 7}\).
3. Not recognizing the need to test boundary and representative values
Students may not realize they need to systematically test values from different parts of the allowed range to see if answer choices can be violated. They might just pick one random value or only test values that seem "convenient" rather than strategically choosing test cases.
1. Testing insufficient or non-strategic example values
When checking answer choices, students might only test values near the middle of the range (like \(\mathrm{x = 2}\)) and miss counterexamples at the edges. For instance, failing to test \(\mathrm{x = 6}\) would prevent them from seeing that Choice C (\(\mathrm{x < 2}\)) can be false.
2. Incorrectly concluding that a choice is always true based on limited testing
Students might test one or two values, see that a statement works for those cases, and incorrectly conclude it must always be true without testing values from across the entire range. This is particularly likely with Choice D where testing only negative values might seem to confirm it.
3. Confusion with inequality directions and transitivity
When working with the logical reasoning that "if \(\mathrm{x > -2}\) and \(\mathrm{-2 > -7}\), then \(\mathrm{x > -7}\)," students might get confused about the direction of inequalities or fail to apply the transitive property correctly, especially when dealing with negative numbers.
1. Settling for the first choice that seems to work
Students might find that Choice B works for a few test values and immediately select it without thoroughly checking all other choices to ensure they can indeed be eliminated. This prevents them from being confident in their answer.
2. Defaulting to "none of the above" when confused
When students feel uncertain about their analysis or have made errors in testing the choices, they might incorrectly choose Option E (none of the above) as a "safe" option, rather than working through their logic more carefully to identify the correct answer.