Problem 63 - Algebra for College Students

Question

Convert each rational expression into an equivalent rational expression that has the indicated denominator. $$\frac{7}{x-1}, \frac{?}{1-x}$$

Step-by-Step Solution

Answer

\(\frac{7}{x-1} = \frac{-7}{1-x} \)

Step 1: Identify the Original Rational Expression

The original rational expression given is \(\frac{7}{x-1}\).

Step 2: Understand the New Denominator

The new denominator we need is \(1-x\).

Step 3: Recognize the Relationship Between Denominators

Notice that \(1-x\) is equivalent to \(-(x-1)\). Therefore, \( \frac{1}{1-x} = -\frac{1}{x-1} \).

Step 4: Convert the Expression

We need to adjust the numerator to compensate for this change. Multiplying both the numerator and the denominator of the original fraction by \(-1\), we get: $$ \frac{7}{x-1} = \frac{7 \times -1}{(x-1) \times -1} = \frac{-7}{1-x} $$.

Key Concepts

Denominator

In rational expressions, the denominator is the bottom part of the fraction. It's what you divide by. For example, in the fraction \(\frac{7}{x-1}\), \(x-1\) is the denominator. The denominator cannot be zero because division by zero is undefined. When working with algebraic denominators, it's important to keep track of changes you make to ensure the fraction remains valid. In the provided exercise, we needed to convert the expression's denominator from \(x-1\) to \(1-x\). This required us to recognize that \(1-x\) is the negative of \(x-1\). Hence, these denominators have a specific relationship that we used to find the equivalent rational expression.

Numerator

The numerator is the top part of a fraction. It tells you how many parts you have. In the fraction \(\frac{7}{x-1}\), \(7\) is the numerator. When adjusting the denominator, you often need to adjust the numerator accordingly. This is important to keep the value of the fraction the same. In our task, to convert \(\frac{7}{x-1}\) into a fraction with the denominator of \(1-x\), we also had to manipulate the numerator. By multiplying the numerator by \(-1\), we got an equivalent expression: \(\frac{-7}{1-x}\). This step ensures the fraction maintains its value despite changing its form.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. For instance, \(\frac{2}{4}\) and \(\frac{1}{2}\) are equivalent because they simplify to the same value. In algebra, you can create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same non-zero number. In our example, we started with \(\frac{7}{x-1}\). We needed a fraction with the same value but a denominator of \(1-x\). By recognizing that \(-1\) times \(x-1\) gives \(1-x\), and similarly adjusting the numerator, we found the equivalent fraction \(\frac{-7}{1-x}\).

Algebraic Manipulation

Algebraic manipulation involves changing the form of an expression using algebraic rules while keeping its original value. This can include multiplying, factoring, expanding, and combining like terms. In our exercise, we performed algebraic manipulation to convert \(\frac{7}{x-1}\) to a form with \(1-x\) as the denominator. First, we identified that \(1-x\) is equal to \(-1 \times (x-1)\). Then, we multiplied the numerator and the denominator of the original fraction by \(-1\), which is a valid operation because multiplying by \(-1\) is akin to multiplying by \(\frac{-1}{-1}\), which equals 1. This gave us the new expression \(\frac{-7}{1-x}\).