How many integer values of x do NOT satisfy (x–4)(x+3)/(x+4)(x+5) > 0?

I think answer is clearly 8 not 10 because , for -4 and -5 , if you substitute in equation we get ( something/0 ) , any mathematic person can say this will tends to infinity , (for ex 1/0 = tends to infinity by concept of limits) , so LHS > 0 is sustained , so answer should be only 8 i.e -3,-2,-1,0,1,2,3,4 .
Bunuel i thinnk OA should be fixed.

We would like to scrutinize the range \(x <= 4\) and \(x >= -5\). When \(x > 4\), we have all four factors as positive and the fraction must be greater than 0. When we have \(x < -5\), all four factors are negative and the fraction must be greater than 0.

Then our focus is the range \(-5 <= x <= 4\), the four values x = 4, -3, -4, -5 either make the fraction 0 or create the divide by 0 error. The values of x between (x = 4) and (x = -3), \(-2 <= x <= 3\) give 3 positive terms and 1 negative term, making the fraction negative so we include all those values too.

In total, we have 4 + 6 = 10 values that do not satisfy this inequality.

Ans: B

multiply both sides by (x+4)^2 (x+5)^2
u get (x-4)(x+3)(x+4)(x+5) > 0
using the number line method. We can identify that -2,-1,0,1,2,3 all give negative values hence less than 0.
Also -4,-3,-5 and 4 makes the equation equal to 0.
so total is 10 values.
IMO B

https://gmatclub.com/forum/solving-ineq ... 74110.html

B should be the answer (10 values) ranging from -5 to +4 (inclusive)
Bunuel can u please clarify?

We see that if x is an integer greater than 4, all the factors are positive; therefore, the expression is positive and hence greater than 0. Similarly, if x is an integer less than -5, all the factors are negative; therefore, the expression is positive and hence greater than 0. Therefore, all the values of x that do not satisfy the inequality must be between -5 and 4, inclusive.

Even though -4 and -5 make the denominator 0, they are still values of x that do not allow the given inequality to hold as true. Thus, these two values are counted.. Furthermore, if x is 4 or -3, we see that the numerator will be 0, which makes the entire expression 0. Since 0 is not greater than 0, we will include 4 and -3 as two additional values of x that do not allow the given equality to hold as true.

Finally, we see that if x is any integer value from -2 to 3, inclusive, (x - 4) will be negative, while the other 3 factors, (x + 3), (x + 4) and (x + 5) will be positive. Therefore, the value of the expression is negative, and hence it’s not greater than 0 for any of these six values of x.

Therefore, there are 2 + 2 + 6 = 10 integer values of x that do not satisfy the inequality. They include all integers between -5 and 4, inclusive.

Answer: B

How can we consider if the value is undefined?

VeritasKarishma Can you please help me to understand this question ?

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