Problem 23 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x+1}{3} \div \frac{3 x+3}{7}$$

Step-by-Step Solution

Answer

The simplified result of the given expression is \( \frac{7}{9} \).

Step 1: Write the division as multiplication

The division of rational expressions can be written as a multiplication by swapping the divisor with its reciprocal. Hence the problem \( \frac{x+1}{3} \div \frac{3 x+3}{7} \) becomes \( \frac{x+1}{3} \times \frac{7}{3x+3} \).

Step 2: Simplify the expression

Before multiplying, look for common factors that can be cancelled in both the numerators and denominators. Factor out like terms for simplification. \(3x+3\) can be rewritten as \(3(x+1)\). So the multiplication becomes \( \frac{x+1}{3} \times \frac{7}{3(x+1)} \). Notice here, \(x+1\) is a common term that can be cancelled.

Step 3: Cancel the common factors

When you cancel the \(x+1\) in both the numerator and the denominator, you have \( \frac{1}{3} \times \frac{7}{3} \).

Step 4: Perform multiplication

Multiply the remaining numerators together to get the numerator of the answer and do the same for the denominators. \( \frac{1}{3} \times \frac{7}{3} = \frac{1 \times 7}{3 \times 3} = \frac{7}{9} \).