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Not understanding few topics in GMAT Club math book

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Sunil01
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Not understanding few topics in GMAT Club math book [#permalink] New post   17 Jan 2016, 00:12
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x = 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01



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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   17 Jan 2016, 00:37
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Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x <= -2 or x >= 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01


Let me try to answer.

Couple of formulae you must know:

1. \(x^2-y^2= (x+y)(x-y)\) . Thus \(x^2-4 = x^2-2^2 = (x+2)(x-2)\)

2. Roots of the inequality\((x-a)(x-b)(x-c)... \geq 0\)are at x=a,b,c... By 'roots' I mean the points at which the expression (x-a)(x-b)(x-c) will change signs when you plot this over the number line.

Coming back to your question,

\(x^2-4 \geq 0\) ---> \(x^2-2^2 \geq 0\) ---> \((x+2)(x-2) \geq 0\) with roots at x=2 and -2.

When you plot the above inequality, you get the following

Attachment:
2016-01-17_2-34-24.jpg
2016-01-17_2-34-24.jpg [ 3.97 KiB | Viewed 1376 times ]


You can clearly see that the region you are looking is the '+' region given by, \(x \leq 2\) and\(x \geq 2\)

Hope this helps.
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Sunil01
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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   17 Jan 2016, 01:02
Engr2012 wrote:
Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x <= -2 or x >= 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01


Let me try to answer.

Couple of formulae you must know:

1. \(x^2-y^2= (x+y)(x-y)\) . Thus \(x^2-4 = x^2-2^2 = (x+2)(x-2)\)

2. Roots of the inequality\((x-a)(x-b)(x-c)... \geq 0\)are at x=a,b,c... By 'roots' I mean the points at which the expression (x-a)(x-b)(x-c) will change signs when you plot this over the number line.

Coming back to your question,

\(x^2-4 \geq 0\) ---> \(x^2-2^2 \geq 0\) ---> \((x+2)(x-2) \geq 0\) with roots at x=2 and -2.

When you plot the above inequality, you get the following

Attachment:
The attachment 2016-01-17_2-34-24.jpg is no longer available


You can clearly see that the region you are looking is the '+' region given by, \(x \leq 2\) and\(x \geq 2\)

Hope this helps.


Hi,

I got till how you calculate the roots i.e 2 and -2.
But how you marked the region on the number line.
Why the region drawn by me as attached in the screenshot is wrong.
if (x+2)(x-2)>=0 then (x+2) >= 0 and (x-2) >= 0
i.e. x >= -2 and x >= 2

This is wrong, but i am not getting where I am going wrong.

Thanks & regards,
Sunil01
Attachments

Absolute values wrong.png
Absolute values wrong.png [ 1.86 KiB | Viewed 1382 times ]

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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   17 Jan 2016, 01:09
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Sunil01 wrote:
Engr2012 wrote:
Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x <= -2 or x >= 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01


Let me try to answer.

Couple of formulae you must know:

1. \(x^2-y^2= (x+y)(x-y)\) . Thus \(x^2-4 = x^2-2^2 = (x+2)(x-2)\)

2. Roots of the inequality\((x-a)(x-b)(x-c)... \geq 0\)are at x=a,b,c... By 'roots' I mean the points at which the expression (x-a)(x-b)(x-c) will change signs when you plot this over the number line.

Coming back to your question,

\(x^2-4 \geq 0\) ---> \(x^2-2^2 \geq 0\) ---> \((x+2)(x-2) \geq 0\) with roots at x=2 and -2.

When you plot the above inequality, you get the following

Attachment:
2016-01-17_2-34-24.jpg


You can clearly see that the region you are looking is the '+' region given by, \(x \leq 2\) and\(x \geq 2\)

Hope this helps.


Hi,

I got till how you calculate the roots i.e 2 and -2.
But how you marked the region on the number line.
if (x+2)(x-2)>=0 then (x+2) >= 0 and (x-2) >= 0
i.e. x >= -2 and x >= 2

This is wrong, but i am not getting where I am going wrong.

Thanks & regards,
Sunil01


You need to review basics of inequality manipulation.

Once you get (x+2)(x-2) >= 0, you do realize that it is of the form a*b >=0 ----> 2 cases

1. Both a,b are >=0 giving you x>=2 or

2. Both a,b <=0 , giving you x<=-2

Thus from 1 and 2 above, you get x<=-2 and x>=2.

Alternately, in order to verify the signs that I mentioned in the attached picture, you can see that for any value of x > 2 , you get (x+2)(x-2) > 0, thus + sign is correct. Similarly for x=-3 you will see that you get a + value.

Hope this helps.
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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   17 Jan 2016, 01:33
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Expert Reply
Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x <= -2 or x >= 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01


Hi Sunil01,

I'll try to explain your query..
\(([square]x[/square] - 4) >= 0\)...
\(([square]x[/square] - 4) = (x-2)(x+2)\)...
so \((x-2)(x+2) >= 0\)...
Now for \((x-2)(x+2) >= 0\), both x-2 and x+2 have to be of same sign..

any value between 2 and -2 will make one of them +ive and the other -ive..
so our answer within this range will be
(x-2)(x+2) < 0...

and x>2 will make both positive and x<-2 will make both negative, making the product >0..

Hope it helps
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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   17 Jan 2016, 20:50
Expert Reply
Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x <= -2 or x >= 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01



Hey Sunil,

These posts discuss the "inequalities with multiple factors" concept in detail. Check them out:

https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/06 ... e-factors/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/07 ... ns-part-i/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/07 ... s-part-ii/
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Sunil01
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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   18 Jan 2016, 05:50
VeritasPrepKarishma wrote:
Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x <= -2 or x >= 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01



Hey Sunil,

These posts discuss the "inequalities with multiple factors" concept in detail. Check them out:

https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/06 ... e-factors/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/07 ... ns-part-i/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/07 ... s-part-ii/



Hi Karishma,

Thanks for sharing the links.
They were an eye opener.
Specially explanation using number line was really amazing.

Regards,
Sunil01
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Re: Not understanding few topics in GMAT Club math book [#permalink] New post   18 Jan 2016, 20:39
Expert Reply
Sunil01 wrote:
VeritasPrepKarishma wrote:
Sunil01 wrote:
Hi Bunuel,

I am reading GMAT Club math book.
I didn't understand how
([square]x[/square] - 4) >= 0 implies x = 2

Please help.
Attached the screenshot also.

Thanks & regards,
Sunil01



Hey Sunil,

These posts discuss the "inequalities with multiple factors" concept in detail. Check them out:

https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/0 ... e-factors/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/0 ... ns-part-i/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/0 ... s-part-ii/



Hi Karishma,

Thanks for sharing the links.
They were an eye opener.
Specially explanation using number line was really amazing.

Regards,
Sunil01


Great they helped you Sunil. You should be able to solve such questions easily now.



Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
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Re: Not understanding few topics in GMAT Club math book [#permalink]
18 Jan 2016, 20:39
Moderator:
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