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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Absolute value & Inequality question
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DavidG@VeritasPrep
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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 CMo2men 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
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Jay@ManhattanReview
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The inequality 5 ≥ |x| ≥ 0 implies either 5 ≥ x ≥ 0 OR -5 ≤ x ≤ 0.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
=> -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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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
GMAT assassins aren't born, they're made,
Rich
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
GMAT assassins aren't born, they're made,
Rich
