Difference between revisions of "1951 AHSME Problems/Problem 21"

Problem

Given: $x > 0, y > 0, x > y$ and $z\ne 0$. The inequality which is not always correct is:

$\textbf{(A)}\ x + z > y + z \qquad\textbf{(B)}\ x - z > y - z \qquad\textbf{(C)}\ xz > yz$ $\textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$

Solution

$\textbf{(A)}\ x + z > y + z\implies x>y$, just subtract $z$ from both sides

$\textbf{(B)}\ x - z > y - z\implies x>y$, just add $z$ to both sides

$\textbf{(C)}\ xz > yz\implies x>y\text{ iff }x>0$, so that means that our desired answer is $\boxed{\textbf{(C)}\ xz > yz}$.

As a check:

$\textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2}\implies x>y$, we can divide $z^2$ safely and without worry because $z^2>0$.

$\textbf{(E)}\ xz^2 > yz^2\implies x>y$, similar reasoning as above but instead, multiply $z^2$.