Problem: The statement x2−x−6<0 is equivalent to the statement:
Answer Choices:
A. −2<x<3
B. x>−2
C. x<3
D. x>3 and x<−2
E. x>3 or x<−2
Solution:
x2−x−6<0,x2−x<6,x2−x+41<6+41,(x−21)2<(25)2
∴∣∣∣∣∣x−21∣∣∣∣∣<25 or −25<x−21<25, that is −2<x<3
or
x2−x−6<0,(x−3)(x+2)<0. This inequality is satisfied if x−3<0 and x+2>0 or if x−3>0 and x+2<0. The first set of inequalities implies −2<x<3; the second set is impossible to satisfy.