Question

Solve the rational equation for x: 3x+x−3x=1x2%0D%0A3%0D%0A𝑥%0D%0A+%0D%0A𝑥%0D%0A−%0D%0A3%0D%0A𝑥%0D%0A=%0D%0A1%0D%0A𝑥%0D%0A2%0D%0A.(1 point)%0D%0AResponses

Answer 1

To solve the rational equation given, we first need to understand it clearly. It seems that the equation is structured as follows:

\[
\frac{3x + x - 3x}{3} = \frac{1}{x^2}
\]

Let's simplify the left-hand side:

1. Combine the terms in the numerator:
\[
3x + x - 3x = x
\]

Thus, we rewrite the left side:
\[
\frac{x}{3} = \frac{1}{x^2}
\]

Next, let's cross-multiply to eliminate the fractions:
\[
x \cdot x^2 = 3 \cdot 1
\]
This simplifies to:
\[
x^3 = 3
\]

Now, we take the cube root of both sides:
\[
x = \sqrt[3]{3}
\]

Thus, the solution to the equation is:
\[
x = 3^{1/3}
\]

It's always good to check for extraneous solutions, especially when dealing with rational equations. In this case, \(x = \sqrt[3]{3}\) is defined and does not make any denominators zero in the original equation.

Therefore, the final solution is:
\[
\boxed{\sqrt[3]{3}}
\]