## Proof That Taking The Reciprocal Reverses the Direction of the Inequality For Positive Numbers

\[x \gt y \gt 0\]

is \[\frac{1}{y} \gt \frac{1}{x} \gt 0\]

? Yes it is.

To prove it divide by

\[x\]

and \[y\]

\[x \gt y \gt 0\]

\[\frac{x}{x} \gt \frac{y}{x} \gt 0\]

\[1 \gt \frac{y}{x} \gt 0\]

\[\frac{1}{y} \gt \frac{1}{x} \gt 0\]

This is only the case for

\[x \gt y \gt 0\]

since multiplying or dividing by a negative number changes the direction of the inequality.