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

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Math Expert
Joined: 02 Sep 2009
Posts: 43804
If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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02 Jan 2017, 04:26
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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
[Reveal] Spoiler: OA

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Posts: 962
If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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02 Jan 2017, 09:44
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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

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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02 Jan 2017, 11:12
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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

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

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

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21 Jan 2017, 16:59
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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
Joined: 02 Sep 2009
Posts: 43804
Re: If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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22 Jan 2017, 03:32
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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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22 May 2017, 19:31
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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22 May 2017, 22:11
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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Posts: 62
If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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23 May 2017, 11:20
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
Joined: 02 Sep 2009
Posts: 43804
Re: If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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23 May 2017, 11:28
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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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23 May 2017, 12:04
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.

Last edited by CantDropThisTime on 23 May 2017, 12:13, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 43804
Re: If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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23 May 2017, 12:11
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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23 May 2017, 12:38
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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31 May 2017, 02:08
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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Joined: 02 Feb 2017
Posts: 17
Concentration: Entrepreneurship, Strategy
Re: If 5 ≥ |x| ≥ 0, which of the following must be true? [#permalink]

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05 Jun 2017, 05:36
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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21 Jan 2018, 12:33
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.

[Reveal] Spoiler:
C

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Re: If 5 ≥ |x| ≥ 0, which of the following must be true?   [#permalink] 21 Jan 2018, 12:33
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