select the correct answer.

which statement describes the solutions of this equation?
\\frac{x}{x+2} + \frac{1}{x} = 1\

- the equation has two valid solutions and no extraneous solutions.
- the equation has no valid solutions and two extraneous solutions.
- the equation has one valid solution and no extraneous solutions.
- the equation has one valid solution and one extraneous solution.

Question

select the correct answer.

which statement describes the solutions of this equation?
\\frac{x}{x+2} + \frac{1}{x} = 1\

- the equation has two valid solutions and no extraneous solutions.
- the equation has no valid solutions and two extraneous solutions.
- the equation has one valid solution and no extraneous solutions.
- the equation has one valid solution and one extraneous solution.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Equations", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Rational Equations", "Extraneous Solutions", "Quadratic Formula Application" ] } ## Identify domain restrictions and clear fractions $$ \begin{aligned} &\text{Equation: } \frac{x}{x+2} + \frac{1}{x} = 1 \ &\text{Restrictions: } x eq -2, \quad x eq 0 \ &\text{Multiply by LCD } x(x+2): \ &x^2 + (x+2) = x(x+2) \end{aligned} $$ ## Solve the resulting quadratic equation $$ \begin{aligned} &x^2 + x + 2 = x^2 + 2x \ &x + 2 = 2x \ &x = 2 \end{aligned} $$ ## Analyze the solution for validity $$ \begin{aligned} &x = 2 \text{ is not equal to } -2 \text{ or } 0. \ &\text{Therefore, } x = 2 \text{ is a valid solution.} \ &\text{There are no other candidate solutions, so there are no extraneous solutions.} \end{aligned} $$

The equation has two valid solutions and no extraneous solutions.

The equation has no valid solutions and two extraneous solutions.

The equation has one valid solution and no extraneous solutions.

The equation has one valid solution and one extraneous solution.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Rational Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify domain restrictions and clear fractions

Solve the resulting quadratic equation

Analyze the solution for validity

Answer: