Problem 7 - Introductory Algebra for College Students

Question

Multiply as indicated. $$\frac{x-3}{x+5} \cdot \frac{4 x+20}{9 x-27}$$

Step-by-Step Solution

Answer

The solution to the multiplication of the two expressions is \( \frac{4}{9} \).

Step 1: Factorize the expressions

Rewrite the expressions by factorizing the polynomials where possible. This includes both the numerator and denominator in each expression in the multiplication. The first expression can't be factorized any further, but the second one can: \( \frac{4x+20}{9x-27} = \frac{4(x+5)}{9(x-3)} \)

Step 2: Multiply the expressions

After step 1, the multiplication becomes: \( \frac{x-3}{x+5} \cdot \frac{4(x+5)}{9(x-3)} \). Simplify by cross-canceling the terms that appear in both the numerator and the denominator: \( (x-3) \) and \( (x+5) \) which will then yield: \( \frac{4}{9} \)

Step 3: Verify the final expression

Ensure that your final expression is the simplest. Here there is no further simplification possible, therefore the simplified multiplication of the given rational expressions is \( \frac{4}{9} \)