Problem 62 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{-1}{x-1} \cdot \frac{1-x}{2}$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1 - Rewrite the Fractions

Rewrite the fractions to handle the sign. Notice \((1-x)\text{ is equivalent to }\text{-(x-1)}\). So the problem becomes: \(\frac{-1}{x-1} \times \frac{-(x-1)}{2} \).

Step 2 - Simplify the Fractions

Simplify the fractions by cancelling out \((x-1) \) in the numerator and denominator: \(\frac{-1}{x-1} \times \frac{-(x-1)}{2} = \frac{-1}{x-1} \times \frac{-1(x-1)}{2} = \frac{-1}{x-1} \times \frac{-(x-1)}{2}\).

Step 3 - Simplify the Expression

Multiplying both fractions, notice that \(-(x-1)\text{ in numerator and denominator cancel each other out: }\frac{-1}{x-1} \times \frac{-(x-1)}{2} = \frac{-1 \times (-1)}{2} = \frac{1}{2}\).

Step 4 - Write the Final Answer

The final answer is \(\frac{1}{2}\).

Key Concepts

Simplifying Algebraic Fractions

Simplifying algebraic fractions is an essential skill in algebra. When we simplify an algebraic fraction, we make it as simple as possible by reducing the expression. This includes factoring both the numerator (top part) and the denominator (bottom part) and canceling common terms.

For instance, consider the exercise: \( \frac{-1}{x-1} \times \frac{1-x}{2} \). To simplify, we first notice that \(1 - x = -(x-1)\). Rewriting the fraction gives us: \( \frac{-1}{x-1} \times \frac{-(x-1)}{2} \).

This equivalent form helps us see common factors in the numerator and denominator. We can cancel these common factors to simplify the expression.

Numerator and Denominator

Understanding numerators and denominators is vital for working with fractions.

The numerator is the top number in a fraction, and it represents the part of the whole.

The denominator is the bottom number and shows how many parts make up the whole.

For example, in the fraction \( \frac{-1}{x-1} \), \(-1\) is the numerator, and \(x-1\) is the denominator.

When multiplying two fractions, such as \( \frac{-1}{x-1} \times \frac{1-x}{2} \), it is essential to identify and manipulate the numerators and denominators correctly to simplify the expression.

Cancelling Terms

Cancelling terms occurs when a factor in the numerator matches a factor in the denominator. This action simplifies the fraction.

In our exercise: \( \frac{-1}{x-1} \times \frac{-(x-1)}{2} \), we have a common factor of \( (x-1) \).

By canceling \( (x-1)\), we simplify the product of the fractions: \( \frac{-1}{x-1} \times \frac{-(x-1)}{2} \).

After canceling \( (x-1) \), the expression simplifies further because \( -1 \) terms multiply to \(1\), leaving \( \frac{1}{2} \) as the final answer. Canceling terms correctly is fundamental to simplifying fractions and solving algebraic problems efficiently.