Question

Select the correct answer. Which statement describes the solutions of this equation? A. The equation has two valid solutions and no extraneous solutions. B. The equation has no valid solutions and two extraneous solutions. C. The equation has one valid solution and no extraneous solutions. D. The equation has one valid solution and one extraneous solution.

Answer 1

The equation can be solved by taking LCM and cross multiplying the terms.

The statement that describes the solutions of the given equation is:

Option C: The equation has one valid solutions and no extraneous solutions.

Given that:

The equation $\dfrac{x}{x+2} + \dfrac{1}{x} = 1$ has to be solved.

What are extraneous solutions?

Those solution values which we get during process of solving equation which aren't really solutions are called extraneous solutions.

Solving the equation:

$\dfrac{x}{x+2} + \dfrac{1}{x} = 1\\\\\text{Taking LCM}\\\\\dfrac{x^2 + (x+2)}{x(x+2)} = 1\\\\\text{Cross multiplying}\\\\x^2 + x + 2 = x^2 + 2x\\\\\text{Subtracting\: } x^2 + \text{x from both sides}\\x^2 + x - x^2 - x + 2 = x^2 + 2x - x^2 - x\\2 = x\\x= 2$

Thus, there is only one solution obtained to the given equation.

Verifying its validity:

$\dfrac{x}{x+2} + \dfrac{1}{x} = 1\\\\\dfrac{2}{2+2} + \dfrac{1}{2} = 1\\\\\dfrac{2}{4} + \dfrac{1}{2} = 1\\\\\dfrac{1}{2} + \dfrac{1}{2} = 1\\\\\dfrac{2}{2} = 1\\\\1 = 1$

Thus, the only solution we got is valid solution.

Thus, Option C: The equation has one valid solutions and no extraneous solutions is correct.

Learn more about valid and extraneous solutions here:

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