sohailmbaprep wrote:how many of the integers that satisfy the inequility
(x+2)(x+3)/(x-2) >=0 are less than 5 ?

The expression = 0 when x=-2 or x=-3 (when numerator is zero)

For cases when x-2>0 ( that is to say that X>2)

The expression can be reduced to inequality
(x+2)(x+3)>0
solution will be x<-3 ; x > -2
But we have already taken the condition x>2
Hence it will be for x>2
Integral values less than 5 = 3 and 4 only


For cases when x-2<0 ( that is to say that X<2)

The expression can be reduced to inequality
(x+2)(x+3)< 0
solution will be -3<x<-2
But we have taken the case where X<2
combining two we get -3<x<-2
We see that there are no integers in this region

Total integers found -3, -2, 3,4

Ans = 4 values less than 5

How many of the integers that satisfy the inequality (x+2)(x+3) / x-2 >= 0 are less than 5?

A) 1

B) 2

C) 3

D) 4

E) 5

One approach is to determine the CRITICAL POINTS: the values where the lefthand side is EQUAL TO 0 or is UNDEFINED.
The lefthand side is equal to 0 when x=-2 and x=-3.
The lefthand side is undefined when x=2.

We already know that x=-2 and x=-3 are valid solutions because they are where (x+2)(x+3) / x-2 = 0.
To determine the range where (x+2)(x+3) / x-2 > 0, try one integer value to the left and right of each critical point.

x < -3:
Plugging x=-4 into (x+2)(x+3) / x-2 > 0, we get:
(-4+2)(-4+3)/(-4-2) > 0
2/-6 > 0.
Doesn't work.
This means that no value less than -3 will work.

-3<x<-2:
No integer values in this range.

-2<x<2:
Plugging x=0 into (x+2)(x+3) / x-2 > 0, we get:
(0+2)(0+3)/(0-2) > 0
-3 > 0.
Doesn't work.
This means that no value between -2 and 2 will work.

x>2:
Plugging x=3 into (x+2)(x+3) / x-2 > 0, we get:
(3+2)(3+3)/(3-2) > 0
30 > 0.
This works.
This means that ANY VALUE greater than 2 will work.
There are only two integer values between 2 and 5:
3 and 4.

Thus, there are four integer values less than 5 that satisfy the inequality: -3, -2, 3 and 4.

The correct answer is D.

Other problems solved with the critical point approach:

https://www.beatthegmat.com/inequality-c ... 89518.html

https://www.beatthegmat.com/knewton-q-t89317.html

https://www.beatthegmat.com/which-is-true-t89111.html