#1**+1 **

sqrt{2x+1} + sqrt{x-3} = 3\sqrt x +7

\(\sqrt{2x+1} + \sqrt{x-3} = 3\sqrt x +7 \)

Since we are in the real number plane, x>=3

\(LHS\\ =\sqrt{2x+1} + \sqrt{x-3} \\ <\sqrt{3x}+\sqrt x \\ <(\sqrt3 +1)\sqrt x\\ < 2.8\sqrt x \\~\\ RHS\\ =3\sqrt x +7\\ >3\sqrt x \\ \)

So the LHS is always smaller than \(2.8\sqrt x\)

And the RHS is always bigger than \(3 \sqrt x \)

So they are never equal. there are no solutions.

I will admit that I tried conventional methods to find the intersestions to start with.

When I got nowhere I plotted them using Desmos.

I could see that they were not going to intersect. The LHS was always less than the RHS.

Then I set out to prove it algebraically. (Which was not difficult.)

Melody Jun 1, 2022