To solve the rational equation
\[ \frac{3x + x - 3x}{x} = \frac{1}{x^2} \]
we can simplify the left side. The \(3x\) and \(-3x\) cancel out, leaving us with
\[ \frac{x}{x} = 1 \quad (\text{for } x \neq 0). \]
So, we have:
\[ 1 = \frac{1}{x^2}. \]
Next, we cross-multiply to eliminate the fraction:
\[ x^2 = 1. \]
This leads to:
\[ x^2 - 1 = 0. \]
Factoring gives us:
\[ (x - 1)(x + 1) = 0. \]
Thus, the solutions are:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1, \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1. \]
Now, we need to check for any restrictions that we might have from the original equation. Since we have \(x\) in the denominator, \(x\) cannot be 0.
The solutions we found are \(x = 1\) and \(x = -1\). Thus, the final answer is:
\[ x = \pm 1. \]
So, the correct answer from the provided responses is:
x = ±1
1 answer