Problem 15 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x-2}{3 x+9} \cdot \frac{2 x+6}{2 x-4}$$

Step-by-Step Solution

Answer

The result of the multiplication is \( \frac{1}{3} \).

Step 1: Factorize the Expressions

Factorize every expression. This is done by looking for the greatest common factor (GCF) in each expression:\[\frac{x-2}{3 x+9} \cdot \frac{2 x+6}{2 x-4} = \frac{x-2}{3(x+3)} \cdot \frac{2(x+3)}{2(x-2)}\]

Step 2: Cancel Out Common Factors

Now, cancel out the common factors in the numerators and denominators. In this case, \(x-2\) and \(x+3\) can be cancelled out:\[\frac{x-2}{3(x+3)} \cdot \frac{2(x+3)}{2(x-2)} = \frac{1}{3} \cdot \frac{2}{2} = \frac{1}{3} \cdot 1\]

Step 3: Multiply the Simplified Expressions

After simplifying the expressions, multiply the fractions together. The multiplication of fractions is straightforward, you just multiply the numerators to create a new numerator, and multiply the denominators to create a new denominator:\[\frac{1}{3} \cdot 1 = \frac{1}{3}\]