Questions: Perform the indicated operation of subtraction on the two rational expressions and reduce your answer to lowest terms. 6x/(1-x) - 2x/(x-1)

Solution Steps

To subtract the two rational expressions, we first need to find a common denominator. Notice that \(1-x\) and \(x-1\) are negatives of each other, so we can rewrite one of them to have a common denominator. After rewriting, we can combine the numerators over the common denominator and simplify the resulting expression to its lowest terms.

The given rational expressions are \(\frac{6x}{1-x}\) and \(\frac{2x}{x-1}\). Notice that \(1-x\) and \(x-1\) are negatives of each other. We can rewrite \(\frac{2x}{x-1}\) as \(-\frac{2x}{1-x}\) to have a common denominator of \(1-x\).

With a common denominator, the subtraction becomes: \[ \frac{6x}{1-x} - \left(-\frac{2x}{1-x}\right) = \frac{6x + 2x}{1-x} = \frac{8x}{1-x} \]

The expression \(\frac{8x}{1-x}\) can be simplified by recognizing that \(1-x = -(x-1)\). Thus, the expression becomes: \[ \frac{8x}{1-x} = \frac{8x}{-(x-1)} = -\frac{8x}{x-1} \]

Final Answer