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Find the range of values of x that satisfy the inequality (x-10)^3 (x+

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Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post Updated on: 23 Dec 2018, 03:55
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Wavy Line Method Application - Exercise Question #6


Find the range of values of x that satisfy the inequality \(\frac{(x-10)^3 (x+7)^5}{(x-5)^5 (x-6)^4} > 0\)

A. -7 < x < 5 or x > 10
B. -7 < x < 5
C. x > 10
D. -6 < x < 5 or x > 11
E. -7 < x < 4 or x > 12




Wavy Line Method Application has been explained in detail in the following post:: http://gmatclub.com/forum/wavy-line-method-application-complex-algebraic-inequalities-224319.html


Detailed solution will be posted soon.

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Originally posted by EgmatQuantExpert on 26 Aug 2016, 02:47.
Last edited by Bunuel on 23 Dec 2018, 03:55, edited 1 time in total.
Edited the question.
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 26 Aug 2016, 10:05
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EgmatQuantExpert wrote:
Wavy Line Method Application - Exercise Question #6


Find the range of values of x that satisfy the inequality \(\frac{(x-10)^3 (x+7)^5}{(x-5)^5 (x-6)^4} > 0\)




Wavy Line Method Application has been explained in detail in the following post:: http://gmatclub.com/forum/wavy-line-method-application-complex-algebraic-inequalities-224319.html


Detailed solution will be posted soon.


Inequality given is \(\frac{(x-10)^3 (x+7)^5}{(x-5)^5 (x-6)^4} > 0\)

We will have the Zero points as -7,5,6 and 10.
Since the power of (x-6)^4 is EVEN, we will not move the curve from 6 to the upside of the line.

So, the range of x will be (-7,5) U (10,infinity)

Please correct me if I am missing anything.
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 18 Nov 2016, 03:26
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2
Solution

Hey Everyone,

Please find below the solution of the given problem.

We see that each factor has a power greater than 1.

We just need to keep in mind that the process is exactly the same: The wavy line passes through the zero point if the power of the corresponding factor is odd. The wavy line bounces back from the zero point if the power of the corresponding factor is even.

(x-6) is the only factor with an even power in the given expression and so the wavy line should bounce back at x = 6.

Plotting the zero points and drawing the wavy line:

Image

Required Range:

-7 < x < 5 or x > 10

Correct Answer: Option A
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 24 Apr 2018, 06:52
EgmatQuantExpert wrote:
Solution

Hey Everyone,

Please find below the solution of the given problem.

We see that each factor has a power greater than 1.

We just need to keep in mind that the process is exactly the same: The wavy line passes through the zero point if the power of the corresponding factor is odd. The wavy line bounces back from the zero point if the power of the corresponding factor is even.

(x-6) is the only factor with an even power in the given expression and so the wavy line should bounce back at x = 6.

Plotting the zero points and drawing the wavy line:

Image

Required Range:

-7 < x < 5 or x > 10

Correct Answer: Option A



For x=6 the answer is zero. Sho should the correct answer then be split into:
1) 6 < x < 10
2) 5 < x < 6
3) x < -7
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 24 Apr 2018, 07:29
kritikalal wrote:
EgmatQuantExpert wrote:
Solution

Hey Everyone,

Please find below the solution of the given problem.

We see that each factor has a power greater than 1.

We just need to keep in mind that the process is exactly the same: The wavy line passes through the zero point if the power of the corresponding factor is odd. The wavy line bounces back from the zero point if the power of the corresponding factor is even.

(x-6) is the only factor with an even power in the given expression and so the wavy line should bounce back at x = 6.

Plotting the zero points and drawing the wavy line:

Image

Required Range:

-7 < x < 5 or x > 10

Correct Answer: Option A



For x=6 the answer is zero. Sho should the correct answer then be split into:
1) 6 < x < 10
2) 5 < x < 6
3) x < -7


We have to find (+) not (-), in this case x = 6 has no meaning for us, since range -7 < x < 5 or x > 10 doesn't include x=6.
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 23 Jul 2018, 03:35
EgmatQuantExpert wrote:
Solution

Hey Everyone,

Please find below the solution of the given problem.

We see that each factor has a power greater than 1.

We just need to keep in mind that the process is exactly the same: The wavy line passes through the zero point if the power of the corresponding factor is odd. The wavy line bounces back from the zero point if the power of the corresponding factor is even.

(x-6) is the only factor with an even power in the given expression and so the wavy line should bounce back at x = 6.

Plotting the zero points and drawing the wavy line:

Required Range:

-7 < x < 5 or x > 10

Correct Answer: Option A



Hi Payal,

Can you please explain why the exclusion holes are marked at 10 and -7 instead of 5 and 6(following the same strategy as in question 4 and in concepts for wavy line)?

TIA

Thanks,
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 23 Jul 2018, 04:55
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Hey sakshi.2sep,

We have explained this in detail in our article.
We suggest you go through it once.

Here is the link for the article:
Wavy Line Method Application: http://gmatclub.com/forum/wavy-line-met ... 24319.html

If you still have any doubt, please let us know.
We will be more than happy to resolve your doubt.

Regards,
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 23 Jul 2018, 05:13
Hi Ashuhtosh,

Appreciate your quick reply.

After going through the article once more, I understand why -7 and 10 are excluded from the solution set but still doubtful if 5 and 6 should also be excluded as they are in the denominator and we should never consider cases where the denominator becomes zero. Please explain why did we not exclude 5 and 6?

Thanks and regards,
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 23 Jul 2018, 05:27
Hey sakshi.2sep,

I have 1 question.

Can we multiply both sides of the inequality by \((x-5)^10(x-6)^8\) in the above fraction?
If you do this then inequality will become \({(x-10)^3 (x+7)^5}{(x-5)^5 (x-6)^4} > 0\)

Now, you can solve the inequality.
Hope this helps you.

Regards,
Ashutosh
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 23 Jul 2018, 05:48
Hey Ashutosh,

Thank you. I do get this question in both the forms: denominator and multiplying by square. Just got confused with the holes (exclusions) marked for 2 points(-7,10) of the solution set and NOT marked for other 2 points (5,6) of the solution set in the wavy line representation. However, the answer mentioned as Required Range: -7 < x < 5 or x > 10 has no doubt.

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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 25 Aug 2018, 08:00
hey EgmatQuantExpert,

You have given great explanation and practice problems.
but I am getting totally confused with starting points of drawing curve.
for above problem, I drew curve starting from top left.
and considered positive parts. where answer is different.

this must be something obvious. but pls help me to get there.
thanks in advance!
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 25 Aug 2018, 10:57
manasveek wrote:
hey EgmatQuantExpert,

You have given great explanation and practice problems.
but I am getting totally confused with starting points of drawing curve.
for above problem, I drew curve starting from top left.
and considered positive parts. where answer is different.

this must be something obvious. but pls help me to get there.
thanks in advance!


Hey manasveek,
you need to start drawing the curve from the top right portion.
I am quoting a part of the article that explains the method:

Quote:
How to draw the wavy line?

1. How to start: Once you draw the number line and mark the zero points on the number line, it’s time to draw the wavy line. Start from the top right most portion. Be ready to alternate (or not alternate) the region of the wave based on how many times a point is root to the given expression.

2. How to alternate:
if the power of a term is odd, then the wave simply passes through the corresponding point (root) into the other region (to –ve region if the wave is currently in the positive region and to the +ve region if the wave is currently in the negative region).


I will suggest you go through the whole article: Wavy Line Method Application - Complex Algebraic Inequalities

Also, read our new article about solving inequalities: Solving inequalities- Number Line Method

Image

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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 27 Aug 2018, 23:06
thank you EgmatQuantExpert for such a quick reply.
I tried going through new article and totally understood concept and why is that we start drawing curve from top right.
a very useful article as always by E-gmat.
Thank you very much!


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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+  [#permalink]

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New post 26 Jul 2019, 12:17
EgmatQuantExpert wrote:
Solution

Hey Everyone,

Please find below the solution of the given problem.

We see that each factor has a power greater than 1.

We just need to keep in mind that the process is exactly the same: The wavy line passes through the zero point if the power of the corresponding factor is odd. The wavy line bounces back from the zero point if the power of the corresponding factor is even.

(x-6) is the only factor with an even power in the given expression and so the wavy line should bounce back at x = 6.

Plotting the zero points and drawing the wavy line:

Image

Required Range:

-7 < x < 5 or x > 10

Correct Answer: Option A


Dear Ms. Tandon,

Thank you for all your replies. I've gone through them, but yet I have one question about the inclusion and exclusion of '6' in the solution if the question was presented differently.

\(\frac{(x-10)^3 (x+7)^5 (x-6)^4}{(x-5)^5} > 0\)
\(\frac{(x-10)^3 (x+7)^5 (x-6)^4}{(x-5)^5} >= 0\)

May I know if x=6 will be included in both of the scenarios?
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Re: Find the range of values of x that satisfy the inequality (x-10)^3 (x+   [#permalink] 26 Jul 2019, 12:17
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