have a finite range

A is incorrect because infinitely many X's could satisfy x^2 + 5x > -6
B is incorrect because infinitely many X's could satisfy x+2 > 4 (even without absolute sign). This also means E is incorrect.
C is incorrect because infinitely many x's could satisfy x < 21/6
D by elimination is correct. However, logically you can see that x^2 - 4x < -3, which would be very tough to satisfy outside of a small range, since x^2 would quickly get larger than 4x.

Had a little difficulty understanding what "finite set" meant, at first I thought it meant there was a least one number that didn't satisfy equation, in which case all but (A) were correct...though x=-3 would cause A to be correct as well, because of the non-inclusive > than symbol.

Then realized it meant a set of X bounded on both sides by real numbers...thanks Wikipedia.

cbenk121 wrote:A is incorrect because infinitely many X's could satisfy x^2 + 5x > -6
B is incorrect because infinitely many X's could satisfy x+2 > 4 (even without absolute sign). This also means E is incorrect.
C is incorrect because infinitely many x's could satisfy x < 21/6
D by elimination is correct. However, logically you can see that x^2 - 4x < -3, which would be very tough to satisfy outside of a small range, since x^2 would quickly get larger than 4x.

Had a little difficulty understanding what "finite set" meant, at first I thought it meant there was a least one number that didn't satisfy equation, in which case all but (A) were correct...though x=-3 would cause A to be correct as well, because of the non-inclusive > than symbol.

Then realized it meant a set of X bounded on both sides by real numbers...thanks Wikipedia.

Excellent reasoning!

Your last point is very important as well. There are actually infinite possible values of x that satisfy all of the inequalities, which is why it's important to focus on the exact wording of the question: we don't need a finite number of values, we need a finite range of values, and only D satisfies that requirement.