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

Question

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

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

Detailed solution will be posted soon.

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Official Answer

A

EgmatQuantExpert · Accepted Answer

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:

https://e-gmat.com/blogs/wp-content/uploads/2016/11/S6.jpg

Required Range  :

-7 < x < 5 or x > 10

Correct Answer  : Option A

abhimahna · Answer

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.

kritikalal · Answer

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

Hero8888 · Answer

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.

sakshi.2sep · Answer

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,
Sakshi

EgmatQuantExpert · Answer

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: https://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, Ashutosh e-GMAT

sakshi.2sep · Answer

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,
Sakshi

EgmatQuantExpert · Answer

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

sakshi.2sep · Answer

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.

Regards,
Sakshi

manasveek · Answer

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!

EgmatQuantExpert · Answer

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:

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

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Ability-Quiz-Algebra-e1533292163516.png

manasveek · Answer

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!

Regards,
Manasvee.

kriskramesh · Answer

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

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