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If 5 ≥ |x| ≥ 0, which of the following must be true?

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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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New post 12 Jun 2018, 13:51
Hi sssjav,

From what you describe, the 'issue' seems to be about how you are interpreting the prompt. I'm going to rephrase the prompt a bit - but NOT change the meaning or the question that is asked.

Consider EVERY number from -5 to +5, inclusive (including negatives, 0, fractions, etc.). Those are the ONLY numbers to consider. Now, consider how each of those possible numbers would answer the three questions here:

I. Is EVERY number in that group greater than or equal to 0?

Of course not; none of the negative numbers are greater than 0.

II. Is EVERY number in that group GREATER than -5?

No. There is one value in the group that is NOT (re: -5).

III. When you square EACH number in that group, will you end up with a value that is between -25 and +25, inclusive?

YES. EVERY value in that group, when squared, will result in a number that is between 0 and +25, inclusive (thus, ALL of those results are between -25 and +25, inclusive).

Thus, the only statement that MUST be true is Roman Numeral 3.

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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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New post 12 Jun 2018, 13:53
Hi Mo2men,

IF Roman Numeral 2 was changed to:

"II. x ≥ -5"

Then that statement would also always be true.

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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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New post 12 Jun 2018, 14:26
EMPOWERgmatRichC wrote:
Hi Mo2men,

IF Roman Numeral 2 was changed to:

"II. x ≥ -5"

Then that statement would also always be true.

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Thanks a lot for reply
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If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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New post 12 Jun 2018, 14:49
We can quickly see that statement 1 and 2 are not always true.
Statement 1 - If X is 1 then yes, but if X is 100 then no
Statement 2 - If X is -4 then yes, but if X is 100 then no

Statement 3 is a bit tricky. You can't take the square root of a negative number so dividing the inequalities by 5 will give the following:

5 ≥ X^2/5≥-5

If X=5, then 5^2/5 = 5 which is sufficient
If X= -5, then -5^2/5 = 5 which is sufficient
All values in between 5 to 0 will still be in the range of satisfying the question

*remember - squaring a negative number will give you a positive value

Answer C
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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New post 13 Jun 2018, 07:53
1
CantDropThisTime wrote:
Bunuel 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


Hi

Option "C" simply can't be correct choice here.

Squire of a real number can never be -ve. It can only be possible only if "x" is an imaginary number. However if "x" is an imaginary number then the condition mentioned in the question itself will not hold true.

Hence option "A-None" should be the correct answer.


Since all three portions of 5 ≥ |x| ≥ 0 are nonnegative, we can square the inequality:
\(5^2\) ≥ \(|x|^2\) ≥ \(0^2\)
25 ≥ \(x^2\) ≥ 0

If \(x^2\) is any value in the range above, it will satisfy the inequality in Statement III.
Thus, Statement III must be true.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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New post 13 Jun 2018, 14:35
Mo2men wrote:
EMPOWERgmatRichC wrote:
Hi All,

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:

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Hi Rich,

Based on your solution above, Can Roman II be correct if it says that x ≥ -5 ??


I must say that I have a similar doubt. I thought that on "must to be true" questions, we need to pick those answers which if they are not, it will counter the initial equation.

In this case, we know that -5 <= x <= 5.

I thought that a "must to be true answer" will be anything that will make sure X is inside these constrains, say -3 < x < 3.

Reflecting uppon that, I figured out that an answer that "must to be true" whould hold for any value of x, say -10 < x. This must to be true... x has to be higher than -5, so it has to be higher than -10 as well.

Does this thought make sense? Would an alternative -10 < x be considered a true one?
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?   [#permalink] 06 Aug 2019, 15:28

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