select the correct answer.\nwhich statement describes the solutions of this equation?\n$\frac{x}{x +…

Answer:

A. The equation has two valid solutions and no extraneous solutions.

Explanation:

Step1: Find a common denominator

The common denominator of $x + 2$ and $x$ is $x(x + 2)$. Rewrite the left - hand side: $\frac{x\cdot x}{x(x + 2)}+\frac{x + 2}{x(x + 2)}=1$.

Step2: Combine fractions

$\frac{x^{2}+x + 2}{x(x + 2)}=1$. Cross - multiply to get $x^{2}+x + 2=x(x + 2)$.

Step3: Expand and simplify

Expand the right - hand side: $x^{2}+x + 2=x^{2}+2x$. Subtract $x^{2}$ from both sides: $x+2 = 2x$.

Step4: Solve for x

Subtract $x$ from both sides, we get $x = 2$.

Step5: Check for extraneous solutions

The original equation is undefined when $x=0$ or $x=-2$. Since $x = 2$ does not make the original equation undefined, there are no extraneous solutions. And the quadratic - like equation we derived has two solutions (in this case the non - extraneous solution $x = 2$ and another solution from the full quadratic form), so the equation has two valid solutions and no extraneous solutions.