Problem 76 - College Algebra Essentials

Question

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Step-by-Step Solution

Answer

\(\frac{4x^{2} + 25x + 15 }{(x-1)(x+2)}\)

Step 1: Distribute the first factor

Distribute \(4-\frac{3}{x+2}\) in \(\left(1+\frac{5}{x-1}\right)\). So, \(\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right) = \left(4-\frac{3}{x+2}\right) + \left(\frac{5}{x-1}\right)\left(4-\frac{3}{x+2}\right)

Step 2: Simplify the expression

Simplicity the expression by doing the multiplication: \( \left( 4 - \frac{3}{x+2} \right) = 4 - \frac{3}{x+2} \) and \(\left(\frac{5}{x-1}\right)\left(4-\frac{3}{x+2}\right) = \frac{20}{x-1} - \frac{15}{(x-1)(x+2)} \).

Step 3: Combine alike terms

Next, combine the similar terms together to simplify the equation further: \( 4 - \frac{3}{x+2} + \frac{20}{x-1} - \frac{15}{(x-1)(x+2)} = 4 + \frac{20x+40-3x-6}{(x-1)(x+2)} - \frac{15}{(x-1)(x+2)} \)

Step 4: Further simplification

Convert the expression to a single fraction by merging the fractions and simplifying numerator: \( 4 + \frac{17x+34-15}{(x-1)(x+2)} = 4 + \frac{17x+19}{(x-1)(x+2)} \)

Step 5: Distribute the '4' into the fraction

To further simplify, make the '4' a fraction and add it to \( \frac{17x+19}{(x-1)(x+2)} \). We get \( \frac{4(x-1)(x+2)+17x+19}{(x-1)(x+2)} = \frac{4x^{2} + 8x - 4 + 17x + 19 }{(x-1)(x+2)} \).

Step 6: Simplify the Fraction

Simplify the numerator to get the final answer: \(\frac{4x^{2} + 25x + 15 }{(x-1)(x+2)} \)