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Math Expert V
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If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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

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3
5
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

0 ≤ x ≤ 5 or -5 ≤ x ≤ 0

I. Not always true because x can be negative.
II. Not always true because x can be equal to -5.
III. $$0 ≤ x^2 ≤ 5^2$$ ---> $$0 ≤ x^2 ≤ 25$$
$$x^2$$ is always positive and ≤ 25. True

##### General Discussion
Director  V
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If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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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

I. x could be 4 or -4....hence not must be true
II. x could be -5 or 5 > -5 ........hence not must be true
III. x^2 = 24 then x<5 (either x= -4.9 or x=4.9 say for instance..)..........TRue

Ans C

Originally posted by rohit8865 on 02 Jan 2017, 09:44.
Last edited by rohit8865 on 21 Jan 2017, 19:50, edited 2 times in total.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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1
Can someone please explain me how C is the correct answer. I chose A.

III. 25 ≥ x^2 ≥ –25

How can we apply square root to -25 and simplify it to 5 ≥ x ≥ –5
Math Expert V
Joined: 02 Sep 2009
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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Sanjeetgujrall wrote:
Can someone please explain me how C is the correct answer. I chose A.

III. 25 ≥ x^2 ≥ –25

How can we apply square root to -25 and simplify it to 5 ≥ x ≥ –5

5 ≥ |x| ≥ 0 means that 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5. For any x from these ranges, 25 ≥ x^2 ≥ –25 will be true.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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1
Bunuel wrote:
Sanjeetgujrall wrote:
Can someone please explain me how C is the correct answer. I chose A.

III. 25 ≥ x^2 ≥ –25

How can we apply square root to -25 and simplify it to 5 ≥ x ≥ –5

5 ≥ |x| ≥ 0 means that 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5. For any x from these ranges, 25 ≥ x^2 ≥ –25 will be true.

Hi Bunuel,
But in the term "25 ≥ x^2 ≥ –25" , doesn't the -25 imply that we cann't take the square root of -25 to simplify the inequality. ?
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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2
Given 0<= |x| <= 5
So absolute value of x can go from 0 to 5 (including both)
So range of x becomes: -5 <= x <= 5
x can go from -5 to 5 (including both)

Now lets look at the given statements.

1. x >= 0
Its not necessary, because x can take negative values also from -5 <= x < 0, and still satisfy the given range

2. x > -5
Its not necessary because x can be = -5 also, and still satisfy the given range

3. -25 <= x^2 <= 25
If you pick any value in the range: -5 <= x <= 5, then it will always satisfy 0 <= x^2 <= 25
Which means it will still satisfy -25 <= x^2 <= 25
So this must be true.

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

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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.
Math Expert V
Joined: 02 Sep 2009
Posts: 64318
Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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ravi11 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.

You did not understand the question.

I'll try to explain again:

5 ≥ |x| ≥ 0 means that 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5. For example, x can be, among infinitely many other values, 0.1, 0.7, 1, 1.7, 4, 5 (because 5 ≥ x ≥ 0) as well as x can be -0.008, -0.4, -3.4, -4, -4.6, -5 (because 0 ≥ x ≥ -5). For ANY possible x, so for ANY x from 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5, it will be true to say that 25 ≥ x^2 ≥ –25.
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If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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Bunuel wrote:
ravi11 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.

You did not understand the question.

I'll try to explain again:

5 ≥ |x| ≥ 0 means that 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5. For example, x can be, among infinitely many other values, 0.1, 0.7, 1, 1.7, 4, 5 (because 5 ≥ x ≥ 0) as well as x can be -0.008, -0.4, -3.4, -4, -4.6, -5 (because 0 ≥ x ≥ -5). For ANY possible x, so for ANY x from 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5, it will be true to say that 25 ≥ x^2 ≥ –25.

Thanks Bunuel for putting such a simplified explanation.However, I am still not convinced.

25 ≥ x^2 ≥ –25 means x^2 can be -24, -23, -22,-21 etc as well. is there any number "x" for which x^2 can be -ve value (-25,-24 etc) and satisfy 5 ≥ |x| ≥ 0 as well.

Please let me know if I am missing an important concept here.

Originally posted by CantDropThisTime on 23 May 2017, 12:04.
Last edited by CantDropThisTime on 23 May 2017, 12:13, edited 1 time in total.
Math Expert V
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Posts: 64318
Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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ravi11 wrote:
Thanks Bunuel for putting such a simplified explanation. I am still not convinced.

25 ≥ x^2 ≥ –25 means x^2 can be -24, -23, -22,-21 etc as well. is there any number "x" for which x^2 can be -ve value (-25,-24 etc) and satisfy 5 ≥ |x| ≥ 0 as well.

Please let me know if I am missing an important concept here.

You are missing the point. The question asks: if 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5, then which of the options must be true. So, if we choose any possible x from the given ranges (5 ≥ x ≥ 0 or 0 ≥ x ≥ -5) and substitute into the options, which option will be always true for any of the possible x's. Any possible x will satisfy 25 ≥ x^2 ≥ –25.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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Bunuel wrote:
ravi11 wrote:
Thanks Bunuel for putting such a simplified explanation. I am still not convinced.

25 ≥ x^2 ≥ –25 means x^2 can be -24, -23, -22,-21 etc as well. is there any number "x" for which x^2 can be -ve value (-25,-24 etc) and satisfy 5 ≥ |x| ≥ 0 as well.

Please let me know if I am missing an important concept here.

You are missing the point. The question asks: if 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5, then which of the options must be true. So, if we choose any possible x from the given ranges (5 ≥ x ≥ 0 or 0 ≥ x ≥ -5) and substitute into the options, which option will be always true for any of the possible x's. Any possible x will satisfy 25 ≥ x^2 ≥ –25.

Thanks Bunuel.

I got the point. So basically all the options that have the question condition as subset of range will be valid.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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1
For those that have difficulty with this question, do not underestimate the value of drawing a number line. It actually made the problem seem easy once I was able to visualize it better.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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laxpro2001 wrote:
For those that have difficulty with this question, do not underestimate the value of drawing a number line. It actually made the problem seem easy once I was able to visualize it better.

Just plug any number based on -5≥x≥5. It solved the questions.
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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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.

GMAT assassins aren't born, they're made,
Rich
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GMAT 1: 710 Q49 V38 If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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Bunuel wrote:
Sanjeetgujrall wrote:
Can someone please explain me how C is the correct answer. I chose A.

III. 25 ≥ x^2 ≥ –25

How can we apply square root to -25 and simplify it to 5 ≥ x ≥ –5

5 ≥ |x| ≥ 0 means that 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5. For any x from these ranges, 25 ≥ x^2 ≥ –25 will be true.

I have a doubt here. Square root of -25 is an imaginary number, 5i. However that lies outside the range of values for x ( Which is [-5,5] )
Doesn't that make Statement-III wrong as well?
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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Hi sssjav,

Based on the information we're given at the beginning of the prompt, we know that X can be any value from -5 to +5 INCLUSIVE. That 'restriction' is what we have to work with when trying to determine which of the three Roman Numerals is ALWAYS TRUE.

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 ALWAYS fall into the range provided by Roman Numeral 3 every time, so Roman Numeral 3 IS true.

GMAT assassins aren't born, they're made,
Rich
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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sssjav wrote:
Bunuel wrote:
Sanjeetgujrall wrote:
Can someone please explain me how C is the correct answer. I chose A.

III. 25 ≥ x^2 ≥ –25

How can we apply square root to -25 and simplify it to 5 ≥ x ≥ –5

5 ≥ |x| ≥ 0 means that 5 ≥ x ≥ 0 or 0 ≥ x ≥ -5. For any x from these ranges, 25 ≥ x^2 ≥ –25 will be true.

I have a doubt here. Square root of -25 is an imaginary number, 5i. However that lies outside the range of values for x ( Which is [-5,5] )
Doesn't that make Statement-III wrong as well?

Numbers on the GMAT are real by default (GMAT deals with only real numbers), so no need to consider complex roots.
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GMAT 1: 710 Q49 V38 If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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EMPOWERgmatRichC wrote:
Hi sssjav,

Based on the information we're given at the beginning of the prompt, we know that X can be any value from -5 to +5 INCLUSIVE. That 'restriction' is what we have to work with when trying to determine which of the three Roman Numerals is ALWAYS TRUE.

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 ALWAYS fall into the range provided by Roman Numeral 3 every time, so Roman Numeral 3 IS true.

GMAT assassins aren't born, they're made,
Rich

I agree to the points you've made, but my doubt remains unresolved.
What you're saying is that for the concerned values of x ( [-5,5] ), Statement-III will always be true - Even though the solution set of statement-III alone might include some values other than those with which we are concerned (All As are Bs, but all Bs are not As).

so what you mean to say is that in a Venn-Diagram-Language, the shape/circle representing [-5,5] will lie enclosed within a bigger shape/circle of statement-III. (You can choose to ignore this statement if it it sounds confusing but you understood my point)

However, by the above logic, even statement-I and statement-II will be true for the concerned values of x, even though the solution sets of the statements might include values other than the ones that belong to [-5,5] and/or might not include some of the values from the set [-5,5].

Now the questions asks us which of the given statements are "true" - according to me, there could only be two possible answers : if we go by the above logic, then all the three statements are true, and If we go by the logic that which of the statements truly represent the all the values of x, then none of the statements will be true. (as all of them represent some values which are either more or less than the concerned set)

However, in another case, if it is asked, that which of the given statements will include ALL the concerned values of x, then statement-III is the best option available. I understand that this is what is meant to have been asked from the question. The thing that I need assistance with is understanding the language of the question - and narrow down on the correct meaning of the question. Where exactly did I interpret the question wrongly? Or what is it that I'm missing?

EDIT : Just read bunuel's reply that gmat does not deal with imaginary number values, that clears up my doubt. Thanks to both of you for your resplies!
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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?  [#permalink]

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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.

GMAT assassins aren't born, they're made,
Rich

Hi Rich,

Based on your solution above, Can Roman II be correct if it says that x ≥ -5 ?? Re: If 5 ≥ |x| ≥ 0, which of the following must be true?   [#permalink] 12 Jun 2018, 01:18

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