Find the range of values of x that satisfy the inequality (x^2-4)(x-5)

Question

Wavy Line Method Application - Exercise Question #5

Find the range of values of x that satisfy the inequality  \frac{(x^2-4)}{(x-5)(x^2-9)} < 0

A. x < -3 or 3 < x < 5

B. x < -3 or -2 < x < 2

C. -2 < x < 2 or 3 < x < 5

D. x < -3 or -2 < x < 2 or 3 < x < 5

E. x < -3

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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https://e-gmat.com/blogs/wp-content/uploads/2018/08/Free-Session-Algebra-e1533292089856.jpg

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, the given inequality can be written as:

(x+2)(x-2)/(x – 5)(x+3)(x-3)<0

Plotting the zero points and drawing the wavy line:

https://e-gmat.com/blogs/wp-content/uploads/2016/11/S5.jpg

Required Range:

x < -3 or -2 < x < 2 or 3 < x < 5

Correct Answer:   Option D

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

abhimahna · Answer

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

it can be written as  \frac{(x-2)(x+2)}{(x-5)(x-3)(x+3)} < 0

So, we have the Zero pints of x as -3,-2,2,3 and 5

But We cannot have x = -3,3 or 5.

So, when drawing these values on the number line, we will get the range of x as

(-infinity,-3)U(-2,2)U(3,5)

Please correct me if I am missing anything.

stonecold · Answer

Here is what i did in this question==>

Critical points => Equate the numerator and denominator terms to zero and obtain the critical points.

In here => x^2-4=0 => x=2 and x=-2
Also x-5=0 => x=5
And finally x^2-9=0=> x=3 and x=-3

Critical points => -3,-2,2,3,5

Now mark these on the number line and pick up a number in each boundary.
If the value makes the original inequality true => Pick that boundary else discard

=> x=> (-∞,-3)U(-2,2)U(3,5)

Personal Opinion-> This question might be good for practise but is most certainly not a GMAT like Question.

The wavy figure that is attached above makes it look scary.In reality we don't need that crazy figure.
Just plug in numbers in a simple plain straight line by marking the critical points.

Shiv2016 · Answer

Does the range include 2 and -2 because the points are not circled on the number line like other points?

shoumkrish · Answer

The range would have included any of the critical points -3, -2, 2, 3 and 5 if the equation read  \frac{(x^2-4)}{(x-5)(x^2-9)} <= 0  (notice the equal sign at the right)

So, to answer your question, it doesn't include 2 and -2.

mt7777 · Answer

Hi, since we have 5 zero points and no number with an even power, can we deduce that the must be 3 or 4 intervals for x? Since there is no answer with 4 intervals and only one with 3 intervals, this must be the right answer, true?

arorni · Answer

Can you please explain, how we have come to the required range?

mt7777 · Answer

This is exactly what she did

I think you have to specify your question a bit. Which part did you not understand?

arorni · Answer

I saw Payal's other post on wavy line https://gmatclub.com/forum/wavy-line-me ... 24319.html. It indeed cleared my doubt about this question, but I am a little confused about drawing wavy lines. Do we always have to start drawing the wavy line from the -ve region (ie Is it the upper right corner)? How the wavy line of (x-1)^2 will look, I got the idea that for even powers the line will remain in the same region but why we have started drawing from the -ve region?

mt7777 · Answer

Ok i got you. As you can see in her approach, the largest zero point for x is 5 (from all the others:-3,-2,2,3,5). So its easy to check if the equation gives a positive or negative result, if you plug in numbers that are less than the smallest zero point (here: -3) or greater the largest zero point (5).

So let's try all number greater the largest zero point, which is 5. For example, plug in  x=10 . Then all terms in the brackets will be positive, so for all  x>5  the equation is  >0 .

Remember: 
 +*+=+ 
 -*+=- 
 -*-=+

Further on, for even powers, the line will not cross the axis, but bounces back, for odd powers it does cross the axis.

bumpbot · Answer

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Find the range of values of x that satisfy the inequality (x^2-4)(x-5)

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Find the range of values of x that satisfy the inequality (x^2-4)(x-5)

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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