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If \(|\mathrm{x}| > 3\), which of the following must be true?
Let's start by understanding what we're being asked. We have a condition that \(|\mathrm{x}| > 3\), and we need to figure out which of three statements (I, II, III) must ALWAYS be true when this condition holds.
The key word here is "must" - we're looking for statements that are guaranteed to be true, not just sometimes true. If we can find even one example where a statement is false while \(|\mathrm{x}| > 3\) is still true, then that statement doesn't "must" be true.
Process Skill: TRANSLATE
Now let's think about what \(|\mathrm{x}| > 3\) actually means in everyday terms. The absolute value \(|\mathrm{x}|\) represents the distance of x from zero on the number line, regardless of direction.
So \(|\mathrm{x}| > 3\) means "x is more than 3 units away from zero." This happens in two situations:
Let's use some concrete examples: \(\mathrm{x} = 4\), \(\mathrm{x} = 10\), \(\mathrm{x} = -4\), \(\mathrm{x} = -10\) all satisfy \(|\mathrm{x}| > 3\).
Mathematically, we can write this as: \(\mathrm{x} > 3\) OR \(\mathrm{x} < -3\)
Process Skill: CONSIDER ALL CASES
Now let's check each statement using our concrete examples:
Statement I: \(\mathrm{x} > 3\)
Let's test with our examples:
Since we found cases where Statement I is false (like \(\mathrm{x} = -4\)), Statement I does NOT must be true.
Statement II: \(\mathrm{x}^2 > 9\)
Let's test with our examples:
Interesting! All our examples work. Let's think about why: if \(|\mathrm{x}| > 3\), then either \(\mathrm{x} > 3\) or \(\mathrm{x} < -3\). In both cases, when we square x, we get \(\mathrm{x}^2 > 9\). This makes sense because squaring eliminates the negative sign.
Statement III: \(|\mathrm{x} - 1| > 2\)
Let's test with our examples:
All our examples work for Statement III too!
We already found that Statement I can be false (when \(\mathrm{x} = -4\), for example), so Statement I is eliminated.
Both Statements II and III worked for all our test cases. Let's verify this more systematically:
For Statement II: If \(|\mathrm{x}| > 3\), then \(\mathrm{x}^2 = |\mathrm{x}|^2 > 3^2 = 9\). This will always be true.
For Statement III: We need to check if \(|\mathrm{x} - 1| > 2\) whenever \(|\mathrm{x}| > 3\).
Both cases work, so Statement III must be true.
Process Skill: APPLY CONSTRAINTS
We determined that:
Therefore, statements II and III only must be true.
The answer is D. II and III only.
1. Misunderstanding "must be true" vs "could be true"
Students often confuse what "must be true" means. They might think a statement is correct if it works for some values (like \(\mathrm{x} = 4\) or \(\mathrm{x} = 10\)) without checking if it fails for other valid values (like \(\mathrm{x} = -4\)). The key insight is that "must be true" means it works for ALL possible values that satisfy \(|\mathrm{x}| > 3\).
2. Incomplete interpretation of absolute value inequality
Many students only consider the positive case when they see \(|\mathrm{x}| > 3\), thinking it just means \(\mathrm{x} > 3\). They forget that absolute value creates TWO conditions: \(\mathrm{x} > 3\) OR \(\mathrm{x} < -3\). This incomplete understanding leads them to miss testing negative values entirely.
3. Not planning to test boundary and edge cases systematically
Students may rush into checking statements without a systematic approach. They might test random values or only positive examples, missing the critical step of testing both positive and negative values that satisfy the constraint.
1. Arithmetic errors when squaring negative numbers
When checking Statement II (\(\mathrm{x}^2 > 9\)), students sometimes make sign errors with negative numbers. For example, they might incorrectly calculate \((-4)^2\) as \(-16\) instead of \(+16\), leading them to wrongly conclude that Statement II doesn't always work.
2. Calculation mistakes with \(|\mathrm{x} - 1|\)
For Statement III, students often make errors when computing \(|\mathrm{x} - 1|\) for negative values. For instance, when \(\mathrm{x} = -4\), they might calculate \(|\mathrm{x} - 1|\) as \(|-4 - 1| = |-3| = 3\), forgetting that \(-4 - 1 = -5\), so the correct calculation is \(|-5| = 5\).
3. Insufficient testing of counterexamples
Students might test only one or two values and conclude prematurely. They may test \(\mathrm{x} = 4\) for all statements, see they work, and assume all statements are true without testing crucial negative values like \(\mathrm{x} = -4\) that would disprove Statement I.
1. Misreading answer choices after correct analysis
Even after correctly determining that only Statements II and III must be true, students might accidentally select "C. I and II only" instead of "D. II and III only" due to rushing or misreading the Roman numerals in the answer choices.
2. Including eliminated statements in final answer
Students may correctly identify that Statement I fails for negative values but then forget this conclusion when selecting the final answer. They might think "Statement II and III work, and Statement I works sometimes" and incorrectly choose "E. I, II, and III" instead of recognizing that "sometimes true" is not the same as "must be true."