Question

Solve the rational equation for x: 3x+x−3x=1x2
.(1 point)
Responses

x=3
x equals 3

x=1
x equals 1

x=0
x equals 0

x=±1
x equals plus or minus 1

Answer 1

To solve the equation \( 3x + x - 3x = \frac{1}{x^2} \), we can first simplify the left side.

The left side simplifies as follows:

\[
3x + x - 3x = x
\]

Now the equation becomes:

\[
x = \frac{1}{x^2}
\]

To eliminate the fraction, we can multiply both sides by \( x^2 \) (assuming \( x \neq 0 \)):

\[
x^3 = 1
\]

Now, we can solve for \( x \) by taking the cube root of both sides:

\[
x = 1
\]

Now, we should also consider the case where \( x = 0 \), since we multiplied by \( x^2 \) which could potentially exclude \( x = 0 \) from the solutions.

When \( x = 0\):

The original equation \( 3x + x - 3x = \frac{1}{x^2} \) has an undefined right side because you cannot divide by zero.

Thus, \( x = 0 \) is not a valid solution.

The only solution we find is:

\[
x = 1
\]

Now, let's summarize the result. The correct answer is:

**x = 1** (or "x equals 1").