If x is a negative integer, what is the value of x?

GMATPrepNow How did you get from: (x+17)(x+20)<0(x+17)(x+20)<0 to -20 < x < -17? My first instinct was to get x<-17 and x<-20?

First notice that (x+17)(x+20) = 0, when x = -17 and when x = -20
Let's examine 3 possible ranges for the value of x.

case i x < -20
If x < -20, then (x+17) is NEGATIVE, and (x+20) is NEGATIVE
So, (x+17)(x+20) = (NEGATIVE)(NEGATIVE) = POSITIVE
So, when x < -20, x is NOT a solution to the inequality (x+17)(x+20) < 0

case ii -20 < x < -17
If -20 < x < -17 then (x+17) is NEGATIVE, and (x+20) is POSITIVE
So, (x+17)(x+20) = (NEGATIVE)(POSITIVE) = NEGATIVE
So, when -20 < x < -17, x IS a solution to the inequality (x+17)(x+20) < 0

case iii x > -17
If x > -17, then (x+17) is POSITIVE, and (x+20) is POSITIVE
So, if (x+17)(x+20) = (POSITIVE)(POSITIVE) = POSITIVE
So, when x > -17, x is NOT a solution to the inequality (x+17)(x+20) < 0

Does that help?

Cheers,
Brent

Statement 1:
If we recognize that x² + 40x + 398 is very close to the PERFECT SQUARE x² + 40x + 400, we can quickly deal with statement 1
Take: x² + 40x + 398 < 0
Add 2 to both sides to get: x² + 40x + 400 < 2
Factor: (x + 20)(x + 20) < 2
In other words: (x + 20)² < 2
Since we're told x is an INTEGER, we can see that there are 3 values of x that satisfy the inequality (x + 20)² < 2
So, x can equal -21, -20 or -19

Statement 2:
Notice that, when -20 < x < -17, we see that (x+17)(x+20) = (NEGATIVE)(POSITIVE) = NEGATIVE
In other words, when x is BETWEEN -20 and -17, (x+17)(x+20) < 0
So, x can equal -18 or -19
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x can equal -21, -20 or -19
Statement 2 tells us that x can equal -18 or -19
Since both statements are TRUE, it must be the case that x = -19
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Posted from my mobile device

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.