Problem: The complete set of x-values satisfying the inequality x2−1x2−4>0 is the set of all x such that:
Answer Choices:
A. x>2 or x<−2 or −1<x<1
B. x>2 or x<−2
C. x>1 or x<−2
D. x>1 or x<−1
E. x is any real number except 1 or -1
Solution:
Method I. Since x2−1x2−4>0,(x2−1>0)⇒(x2−4>0). Therefore, all real values of x such that x>2 or x<−2 satisfy the inequality. Also (x2−1<0)⇒ (x2−4<0). Therefore, all real values of x such that −1<x<1 satisfy the inequality.
Method II. Since x2−1x2−4>0,x2−1x2−1−x2−13>0. Therefore, 1>x2−13.
This latter inequality is satisfied by all x such that x2−1>3, that is, by x>2 or x<−2, and by all x such that x2−1<0, that is, by −1<x<1.