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Find all x that satisfy the equation
sqrt{2x+1} + sqrt{x-3} = 3\sqrt x +7

 Jun 1, 2022
 #1
avatar+117494 
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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.)

 Jun 1, 2022
 #2
avatar+117494 
0

A response is always appreciated

 Jun 1, 2022

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