Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{6}{x-1}-\frac{x}{1-x} $$

Short answer

\[\frac{x + 6}{x-1}\]

Step-by-step solution

Identify the denominators

The denominators in the given fractions are \(x-1\) and \(1-x\). Notice that \(1-x = -(x-1)\).

Rewrite the second fraction

Rewrite \(\frac{x}{1-x}\) as \(\frac{x}{-(x-1)} = -\frac{x}{x-1}\). This step gives us a common denominator. Now the expression is: \[\frac{6}{x-1} - (-\frac{x}{x-1})\].

Combine the fractions

Combine the fractions over a common denominator: \[\frac{6}{x-1} + \frac{x}{x-1} = \frac{6 + x}{x-1}\].

Simplify the numerator

Simplify the numerator by combining the terms: \[\frac{6 + x}{x-1} = \frac{x + 6}{x-1}\].

Ensure the result is in factored form

The numerator \(x + 6\) and the denominator \(x-1\) are already in their simplest factored forms. So the final answer remains: \[\frac{x + 6}{x-1}\].

Key concepts

fractions

Fractions represent parts of a whole number. They consist of two main components: the numerator and the denominator. In the fraction \[ \frac{6}{x-1} \], 6 is the numerator, and \[ x-1 \], the denominator, shows how many parts the whole is divided into. Fractions can be added, subtracted, multiplied, and divided. Each operation follows specific rules, especially when working with algebraic expressions.
Understanding the roles of the numerator and denominator is crucial when manipulating and simplifying algebraic fractions.

common denominator

To add or subtract fractions, they must have a common denominator. This means the denominators of the fractions must be the same. This allows us to combine the numerators easily. In the given problem, the denominators are \[ x-1 \] and \[ 1-x \].
By recognizing that \[ 1-x = -(x-1) \], we rewrite the second fraction to have a common denominator. This results in \[ \frac{6}{x-1} - (-\frac{x}{x-1}) = \frac{6}{x-1} + \frac{x}{x-1}. \] Now, both fractions share the denominator \[ x-1 \], enabling us to combine them.

simplification

Simplification makes algebraic expressions more manageable. It involves combining like terms and reducing fractions to their simplest forms. In our exercise, after rewriting the second fraction, we combine numerators:

• \[ \frac{6 + x}{x-1} \] combines the numerators (6 + x) over the common denominator (x-1).
• This expression is already simplified since no further reduction is possible.

Simplification helps in reducing complexity, making it easier to understand and solve algebraic problems.

factored form

Factored form expresses an algebraic expression as a product of its factors. Factoring is useful for simplifying expressions, solving equations, and finding common denominators. In the problem, the final answer is
\[ \frac{x + 6}{x-1} \]. Both the numerator (x + 6) and the denominator (x-1) are already in their simplest factored forms. Factoring reveals the building blocks of expressions, making operations and further simplification clearer.