Prove that if $x \in \mathbb R$ and $x>2$ then $y+ \frac{1}{y} = x$ will have real solution.

This is essentially correct. One small thing which is missing: you should say that the equation $y^2-xy+1$ has non-zero real solutions, since the step where you multiplied both sides by $y$ could in principle have added an extra "solution" $y=0$ which was not a solution of the original equation. But that has not happened in this case since clearly $y=0$ is not a solution of your quadratic.

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Algebra Precalculus

Proof Verification

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