What is the easiest way to sketch $\frac{1}{x} > \frac{1}{y}$?

Question

When I do it, it never goes anywhere the first time because there are so many conditions and cases when x or y is positive or negative. The worked solutions suggested a "quadrant" based method but it is still confusing. Is there an even easier way or will a question like this naturally be difficult to solve?

Answer 1

• If $xy>0$ (the first and third quadrant excluding the vertical and horizontal axis), multiply $xy$ to both sides of the inequality, we have $y > x$ in the first and the third quadrant.

• If $xy < 0$ (the second and fourth quadrant excluding the vertical and horizontal axis), multiply $xy$ to both sides of the inequality, we have $y < x$. This is just the entire fourth quadrant.

Answer 2

here are the worked solutions (as requested). I guess its relatively concise - is there an even better way?

Answer 3

Multiply both sides of the inequality by $xy$.

If $xy > 0$, this means $x > 0$ and $y > 0$, or $x < 0$ and $y < 0$. We have $y > x$, so sketch the upper half of the 1st quadrant and the upper half of the 3rd quadrant.

If $xy < 0$, this means $x < 0$ and $y > 0$, or $x > 0$ and $y < 0$. This gives $y < x$, but this is violated when $x < 0$ and $y > 0$. Therefore sketch the entire 4th quadrant as both conditions are satisfied.