No takers? No problem, it's a tricky question!


Important: Please note that the question is NOT asking us to determine all possible values of \(x^2\). The question is asking us to determine which of the inequalities MUST be true?

For this question, we must avoid the temptation of squaring all three parts of the inequality to get: \(4 < x^2 \leq {25}\)
A quick way to see the error in this strategy is to ask yourself what your answer would be if it was given that \(-5 < x < {5}\)
Here, if we square all three parts of the inequality, we get: \(25 < x^2 < 25\), which makes no sense.
The main issue is that squaring negative numbers produces positive numbers that reduce the set of possible values of \(x^2\).

For this kind of question, I usually test values. It's fast and helps avoid careless mistakes.

If -2 < x < 5, then it could be the case that x = 5, in which case x² = 25
In other words, it's possible that x² = 25
Check the answer choices...
Answer choice A says x² < 16. In other words, it says that x² can't equal 25. ELIMINATE A
Answer choice C says x² < 16. In other words, it says that x² can't equal 25. ELIMINATE C

Also, if -2 < x < 5, then it could be the case that that x = 0, in which case x² = 0
In other words, it's possible that x² = 0
When we check the remaining answer choices, we see that...
Answer choice B (4 < x²) says x² can't equal 0. ELIMINATE B
Answer choice D (0 < x²) says x² can't equal 0. ELIMINATE D

We're left with 1 answer choice...E

NOTE: Some students will assume that E is incorrect, because x² CAN'T equal -4. However, answer choice E doesn't suggest that x² can equal -4; it only states that x² must be GREATER THAN or equal to -4, which is true.