solving set a set of in-equations for x and y

Question

I have a set of below in-equations. Can we solve for a solution of x and y

$$x>2$$ $$xy>5$$ $$y^2x\leq9$$

Answer 1

If you plot the curves of equality you get a graph like the below. The blue is $xy=5$ and the orange is $xy^2=9$. We want the space on or below the orange, above the blue, and right of $x=2$. We can find the crossing point by solving the equalities simultaneously. $$xy=5\\ xy^2=9 \\ y=\frac 95 \\x=\frac {25}9$$ The infinite region between the curves and right of the crossing point is acceptable.

Answer 2

From the second inequality we find that $y>0$. The last two inequalities implies that $\frac{5}{x}<y\leq \frac{9}{x^2}$, which is possible only when $x\geq\frac{25}{9}$. My idea is from the method of “combination of calculation and graph” which I used a lot in high school. Ross Millikan just took this way to solve your problem. And I really recommend this method.