Problem 33 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{a-b}{2 b-2 a}$$

Step-by-Step Solution

Answer

-\frac{1}{2}

Step 1: Identify Common Factors

First, observe the numerator and the denominator to find any common factors. The numerator is given as \(a - b\), and the denominator is \(2b - 2a\).

Step 2: Factor the Denominator

Factor out the common factor from the denominator. The expression \(2b - 2a\) can be factored as follows: \(2(b - a)\).

Step 3: Rewrite the Expression

Rewrite the original fraction with the factored denominator. The expression now looks like this: \(\frac{a - b}{2(b - a)}\).

Step 4: Simplify the Expression

Notice that \(a - b\) and \(b - a\) are similar but have opposite signs. Therefore, \(b - a = -(a - b)\). Substitute this into the denominator: \(\frac{a - b}{2(-1)(a - b)}\).

Step 5: Reduce the Fraction

Since the \(a - b\) terms in the numerator and denominator are identical, cancel them out: \(\frac{1}{-2}\). The simplified form of the expression is \(-\frac{1}{2}\).

Key Concepts

Common Factors

When simplifying rational expressions, the first step is to identify common factors in both the numerator and the denominator. A common factor is a number or an expression that divides both the numerator and the denominator evenly.

In our original problem, we have the rational expression: \[ \frac{a - b}{2b - 2a} \]

The numerator here is \(a - b\), and the denominator is \(2b - 2a\). To find common factors, look for terms that appear in both parts. Here, we see \(2\) is a common factor in the denominator. Spotting these common factors allows us to move to the next step: factoring.

Factoring

Factoring is the process of breaking down an expression into products of simpler expressions. In this exercise, the denominator needs to be factored to observe further simplification opportunities. Starting with \(2b - 2a\), factor out the greatest common factor (GCF), which is \(2\).

Our expression becomes: \[ 2(b - a) \]

Now, we have rewritten the original fraction: \[ \frac{a - b}{2(b - a)} \]

Factoring helps transform the expression into a simpler form, making the next steps easier. Always look for common factors to factor out, making the expression simpler to handle.

Simplification of Fractions

Simplification is the process of reducing a fraction to its lowest terms. In our case, we need to simplify the expression: \[ \frac{a - b}{2(b - a)} \]

Notice that \(a - b\) and \(b - a\) are similar but not identical. \(b - a\) is the negative of \(a - b\), i.e., \(b - a = -(a - b)\).

Substitute this relation into the denominator to get: \[ \frac{a - b}{2(-1)(a - b)} \]

The \(a - b\) terms in both the numerator and the denominator can now be canceled out: \[ \frac{1}{-2} \]

Thus, the simplified form of the given fraction is: \[ -\frac{1}{2} \]

Every time you simplify a fraction, you should look for common factors and use techniques like factoring and substitution to reduce the expression efficiently.