Find the product of the integer values of x that satisfy the inequalit

Question

Exercise Question #7 of the    Wavy Line Method Application

Find the product of the integer values of x that satisfy the inequality  \frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} > 0

A. -2
B. -1
C. 0
D. 1
E. 2

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Apologies for the inconvenience. Question edited appropriately to correct the earlier typo.

Detailed solution will be posted soon .

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EgmatQuantExpert · Accepted Answer

Solution

Hey Everyone,

Please find below, the solution of the given problem.

Rewriting the inequality to easily identify the zero points

We know that $(x^2 – 4) = (x+2)(x-2)$

And similarly, $(x^2-9) = (x+3)(x-3)$

So, $(x^2-4)^3/((x – 5)^5 (x^2-9)^4 )>0$ can be written as

$((x+2)^3 (x-2)^3)/((x – 5)^5 (x+3)^4 (x-3)^4 )>0$

(x + 3) and (x-3) are the terms with even power. So, the wavy line should bounce back at the corresponding zero-points x = -3 and x = 3.

Plotting the zero points and drawing the wavy line:

Required Range:

-2< x < 2 or x > 5

Calculations:

We need the product of all integers in this range. However, note that the integer 0 is a part of the range (-2 < x < 2).

Therefore, the product will be 0.

Correct Answer: Option C

abhimahna · Answer

The inequality is  \frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} < 0

or  \frac{(x-2)^3 (x+2)^3}{(x-5)^5 (x - 3)^4 (x + 3)^4} < 0

Now, we have (x-3) and (x+3) with EVEN powers, so we will not move the curve direction to the upper side.

So, the zero points of x will be -3,-2,2,3 and 5

But we cannot have x = 5,3 or -3 as they are at the denominator.

So, the range of x will be (-infinity, -2) U (2,5).

Please correct me if I am missing anything.

yezz · Answer

^3 / (x-5)^5 ( x^2-9)^4

critical values are -2,2,5,

the inequality is valid in ranges x<-3 and 2<x<5 , x not equal 3 , thus +ve integer value of x is 4 , and the -ve integer values belong to (-infinity,-4)

how can i get the product then or where am i going wrong??

OreoShake · Answer

just had a question to solidify my understanding of this - if the question was  <  instead of  > , then there would be  no  ranges that satisfy this equation? Thank you.

EgmatQuantExpert · Answer

Hey,

If the question was less than (<) instead of greater than(>), then we  can still find the range of the range of x.

Look at the wavy curve in the solution given above and think, if we had to find the range of x for which the expression is less than zero, what range will we consider?

Hint:   We had considered the positive range in the wavy curve to find the range of x for which the expression was more than zero.

Thanks,
Saquib
Quant Expert
e-GMAT

broall · Answer

We could solve this question easily with the tool called: factor table with sign $\frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} > 0 \iff \frac{(x-2)^3(x+2)^3}{(x-5)^5(x-3)^4(x+3)^4} > 0$ First, find the value that all factors are equal to 0 $\begin{split} x-2=0 &\iff x=2 \ x+2=0 &\iff x=-2\ x-5=0 &\iff x=5\ x-3=0 &\iff x=3\ x+3=0 &\iff x=-3 \end{split}$ Now, make a factor table with sign like this Capture table 1.PNG In line (1), we review all possible values of $x$ that make each fator equal to 0. The order is ascending (from lowest to highest) Next, from line (2) to (6), we review the sign of each raw factor without power that based on the range value of $x$. Character "|" means that we no need to care about the specific value of them, just the sign positive (+) or negative (-). Next, from line (7) to (11), we review the sign of each fator with power that exists in the expression. Finally, in line (12), we could quickly review the sign of the expression. Hence, we have $\frac{(x-2)^3(x+2)^3}{(x-5)^5(x-3)^4(x+3)^4} > 0 \iff x \in (-2,2) \cup (5, +\infty) $ You could find more about this tool in this link: fator-table-with-sign-the-useful-tool-to-solve-polynomial-inequalities-229988.html#p1771368

shanksfire · Answer

Given inequality  \frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} > 0

Since only expressions with odd powers play role in determining the sign of the equation, 
1.) we can remove the even power expressions and 
2.) make all odd powers equivalent to 1.
We also note that x cannot be equal to 5, 3, -3 as these will make the numerator '0'.

Thus, we can solve for below inequality instead.
  \frac{(x^2-4)}{(x-5)} > 0

Or,
  {(x^2-4)(x-5)} > 0

Points are : -2, 2 and 5.
Thus, Ans: (-2,2) U (5,infinity).

KS

soodia · Answer

Hi

for satisfying the inequality (x^2-4)^3 and ((x – 5)^5 should have the same sign
so, they could be both negative or positive
the range will be different
am I wrong?

niks18 · Answer

Hi soodia the range of x is -25 . In this range the expression \frac{(x^2-4)^3}{(x-5)^5} will always be greater than 0 . you can test this by using some values from the range let x=-1 then \frac{(x^2-4)^3}{(x-5)^5} = \frac{(1-4)^3}{(-1-5)^5}>0 if x=6 then \frac{(x^2-4)^3}{(x-5)^5}=\frac{(6^2-4)^3}{(6-5)^5}>0 Hi EgmatQuantExpert we know that (x^2-9)^4>0 and x≠3 , -3 & 5 because in that case denominator will be 0 which is not possible. So isn't our equation be simplified as \frac{(x^2-4)^3}{(x-5)^5}>0 (cross-multiplying (x^2-9)^4 ) And the wavy line for this equation will be as below giving us the same range -25 inequality 1.jpg

4d · Answer

Hi,

Just a heads up .

The solution set is  (-2,2)  U  (5,\infty) 
So the product of all integers  = -1 	imes 0 	imes 1 	imes 2 	imes .... 10 	imes ... \infty 
So basically we are multiplying  ightarrow 0 	imes -\infty = undefined

Its better to tweak the question so as to get a finite integer set . Then the answer will be zero .

Kinshook · Answer

Asked: Find the product of the integer values of x that satisfy the inequality  \frac{(x^2-4)^3}{(x-5)^5 (x^2 - 9)^4} > 0

\frac{(x-2)^3(x+2)^3}{(x-5)^5(x-3)^4(x+3)^4} >0 
if equivalent to: 
 (x-2)(x+2)/(x-5) >0 & x 
eq -3 & x 
eq 3

x = {-1,0,1,6,.....}

Product of all integer values of x = 0*(-1)*1*6*... = 0

IMO C

Noida · Answer

How can we say that infinity multiplied by zero will give zero? Won't that be undefined?

basicallybhavesh · Answer

Any number multiplied by 0 is 0.

346273864782748237490384923480234982342490 * 0 = 0.

bumpbot · Answer

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Find the product of the integer values of x that satisfy the inequalit

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Find the product of the integer values of x that satisfy the inequalit

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