1974 AHSME Problems/Problem 14

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Problem

Which statement is correct?

$\mathrm{(A)\ } \text{If } x<0, \text{then } x^2>x. \qquad \mathrm{(B) \ } \text{If } x^2>0, \text{then } x>0.$

$\qquad \mathrm{(C) \  } \text{If } x^2>x, \text{then } x>0. \qquad \mathrm{(D) \  } \text{If } x^2>x, \text{then } x<0.$

$\qquad \mathrm{(E) \  }\text{If } x<1, \text{then } x^2<x.$

Solution

We begin by going statement by statement.

For (A), $x^2>0$ for all real $x$, and so $x^2>0>x$. Therefore, $\boxed{\text{A}}$ is true.

We can give counterexamples for all other statements:

For (B), $x=-1$

For (C), $x=-1$

For (D), $x=2$

For (E), $x=-1$

See Also

1974 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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