If |x|/|3| > 1, which of the following must be true?

If |x|/|3| > 1, which of the following must be true?

A. x > 3
B. x < 3
C. x = 3
D. x ≠ 3
E. x < -3

Notice that if x = 3, then \(\frac{|x|}{|3|} = \frac{|3|}{|3|} = 1\), so \(\frac{|x|}{|3|}\) is NOT more than 1, it's equal to 1. Thus if x = 3, the given inequality does NOT hold true.

As for the other options.

First, simplify the given inequity: \(\frac{|x|}{|3|} > 1\) --> \(\frac{|x|}{3}> 1\) --> \(|x| > 3\) --> \(x < -3\) or \(x > 3\). This is given as a fact.

Now, if x < -3 or x > 3, then which of the options MUST be true?

A. x > 3 --> this option is not necessarily true since x could be less than -3, for example -4, which will make this options not true.

B. x < 3 --> this option is not necessarily true since x could be more than 3, for example 4, which will make this options not true.

C. x = 3 --> this option is NEVER true since we know that x < -3 or x > 3.

D. x ≠ 3 --> we know that x < -3 or x > 3. Thus x cannot be 3. Thus this option is true.

E. x < -3 --> this option is not necessarily true since x could be more than 3, for example 4, which will make this options not true.

Answer: D.

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Hope it helps.