Solve

Let $u= \sqrt{4x-3}$ . Thus, our equation becomes: $u + \frac{10}{u} = 7$ , or $u^2 - 7u + 10$ . Solving this quadratic gives $u=2$ or $u=5$ .

If $u=2$ , then $\sqrt{4x-3} = 2$ , or $x = \frac{7}{4}$ .

If $u=5$ , then $\sqrt{4x-3} = 5$ , or $x = 7$ .

Since the larger of these two solutions, is $7$ , our answer is $\fbox{7}$ .