Problem 15 - College Algebra

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 multiplying \(\frac{(x-2)}{(3x+9)} \cdot \frac{(2x+6)}{(2x-4)}\) is \(\frac{1}{3}\).

Step 1: Simplify the Fractions

The given expression is \(\frac{(x-2)}{(3x+9)} \cdot \frac{(2x+6)}{(2x-4)}\). Now, let's simplify the fractions. It can be noticed that both denominators can be factored. The fraction \((3x+9)\) can be factored into \(3(x+3)\) and the fraction \((2x+6)\) can be factored into \(2(x+3)\). The original expression now becomes \(\frac{(x-2)}{3(x+3)} \cdot \frac{2(x+3)}{(2x-4)}\).

Step 2: Cancel Common Factors

By the properties of fractions, we see that \((x+3)\) appears in both the numerator and the denominator of the expression, you can cancel out this term in both to further simplify the equation. The expression becomes \(\frac{x-2}{3} \cdot \frac{2}{2x-4}\).

Step 3: Multiplying the simplified fraction

Multiplying these fractions by simply multiplying the numerators together and the denominators together, we get \((x-2)*2/((3)*(2x-4))\), which simplifies down to \(\frac{2(x-2)}{3(2x-4)} = \frac{2x-4}{6x-12}\).

Step 4: Simplify

We can still simplify this fraction. The denominator \(6x-12\) can be factored into \(6(x-2)\) giving us \(\frac{2x-4}{6(x-2)}\). Both the numerator and the denominator share a common factor of \(x-2\) which can be cancelled out. After cancelling out \(x-2\) on top and bottom we are left with \(\frac{2}{6} = \frac{1}{3}\).