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Algebra

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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
+118477
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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.

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
+118477
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A response is always appreciated

Jun 1, 2022