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Absolute value & Inequality question

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by Mo2men » Tue Jan 03, 2017 1:47 pm
If 5 ≥ |x| ≥ 0, which of the following must be true?

I. x ≥ 0
II. x > -5
III. 25 ≥ x^2 ≥ -25

A. None
B. II only
C. III only
D. I and III only
E. II and III only
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by DavidG@VeritasPrep » Tue Jan 03, 2017 2:40 pm
Mo2men wrote:If 5 ≥ |x| ≥ 0, which of the following must be true?

I. x ≥ 0
II. x > -5
III. 25 ≥ x^2 ≥ -25

A. None
B. II only
C. III only
D. I and III only
E. II and III only
Test an extreme value of x. Say x = -5. I and II wouldn't be true, so it's either III alone or none of the above. Notice that no matter what we pick (x = -4 or x = -3, etc) the square of x will always be between 25 and -25 inclusive. (Notice also that x^2 will always be nonnegative. All nonnegative numbers are clearly larger than -25, so this isn't an issue.) The answer is C
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by Jay@ManhattanReview » Tue Jan 03, 2017 8:48 pm
Mo2men wrote:If 5 ≥ |x| ≥ 0, which of the following must be true?

I. x ≥ 0
II. x > -5
III. 25 ≥ x^2 ≥ -25

A. None
B. II only
C. III only
D. I and III only
E. II and III only
The inequality 5 ≥ |x| ≥ 0 implies either 5 ≥ x ≥ 0 OR -5 ≤ x ≤ 0.

=> -5 ≤ x ≤ 5

=> 0 ≤ x^2 ≤ 25; x^2 is always a positive number.

Only statement III must be true. Though x^2 ≥ -25 does not make any sense since x^2 is a non-negative number, 0 ≤ x^2 ≤ 25 lies within 25 ≥ x^2 ≥ -25; statement III is valid.

OA: C

Hope this helps!

-Jay
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by [email protected] » Tue Jan 03, 2017 8:54 pm
Hi Mo2men,

This question can be solved by TESTing VALUES. Notice the specific inequalities that we're given to work with - based on the information in the prompt, we know that X can be any value from -5 to +5 INCLUSIVE. We're asked which of the following MUST be true.

I. x ≥ 0
II. x > -5

For Roman Numerals 1 and 2, you could consider X = -5. With that value, neither of those two Roman Numerals is true.
Eliminate Answers B, D and E.

III. 25 ≥ x^2 ≥ -25

Roman Numeral 3 asks us to think about SQUARED terms. With the given range of values that we have to work with, the range of the squared terms would be 0 through +25, inclusive. Regardless of the exact value that you choose for X, X^2 will fall into the range provided by Roman Numeral 3 every time, so Roman Numeral 3 IS true.

Final Answer: C

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