Existence and uniqueness proof

#existence#uniqueness#proof

Statement

Prove $\exists! x\in\mathbb{R},\ 2x+1=5$ .

Hint

Solve the equation, then show uniqueness.

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Existence. $x=2$ satisfies $2\cdot2+1=5$ .

Uniqueness. If $2x+1=5$ , then $2x=4$ , so $x=2$ . Unique.

Conclusion. $x=2$ is the unique solution.