1969 AHSME Problems/Problem 14

Problem

The complete set of $x$-values satisfying the inequality $\frac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that:

$\text{(A) } x>2 \text{ or } x<-2 \text{ or} -1<x<1\quad  \text{(B) } x>2 \text{ or } x<-2\quad \\ \text{(C) } x>1 \text{ or } x<-2\qquad\qquad\qquad\quad \text{(D) } x>1 \text{ or } x<-1\quad \\ \text{(E) } x \text{ is any real number except 1 or -1}$

Solution

Factor the difference of squares. \[\frac{(x+2)(x-2)}{(x+1)(x-1)}>0\] Note that the graph intersects the x-axis at when $x = \pm2$ or $x \pm 1$, so check the sign of the result to see if it is positive. After testing, $x<-2$ or $-1<x<1$ or $x>2$, so the answer is $\boxed{\textbf{(A)}}$.

See also

1969 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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