Largest possible value of $x$ Problem:

Let $x$ and $y$ be real numbers satisfying $\frac{x^2y^2 - 1}{2y-1}=3x.$ Find the largest possible value of $x.$

How would I do this? Would I multiply each side by $2y-1$?

Hint: eliminate the denominator and consider it as a quadratic equation in $y\,$: $$x^2 \cdot y^2 - 6x \cdot y +3x -1 = 0$$
For the quadratic to have real roots, its discriminant must be non-negative:
$$
\frac{1}{4} \Delta = 9 x^2 -x^2(3x -1) = x^2(10 - 3x) \ge 0
$$
