What is the solution of the equation? $$ \sqrt{2 x-1}+2=x $$ Enter your answer in the box. $$ x=\square $$

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Solve the equation $$ \sqrt{2x-1}+2=x $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\sqrt{2x-1}+2=x$$
- step1: Find the domain:
  $$\sqrt{2x-1}+2=x ,x\geq \frac{1}{2}$$
- step2: Move the constant to the right-hand side:
  $$\sqrt{2x-1}=x-2$$
- step3: Evaluate:
  $$\sqrt{2x-1}=x-2,x-2\geq 0$$
- step4: Evaluate:
  $$\sqrt{2x-1}=x-2,x\geq 2$$
- step5: Solve the equation:
  $$\begin{align}&x=5\&x=1\end{align},x\geq 2$$
- step6: Find the intersection:
  $$x=5$$
- step7: Check if the solution is in the defined range:
  $$x=5,x\geq \frac{1}{2}$$
- step8: Find the intersection:
  $$x=5$$
- step9: Check the solution:
  $$x=5$$
The solution to the equation $$ \sqrt{2x-1}+2=x $$ is $$ x=5 $$.
$$ x=5 $$

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