Multiply or divide as indicated. $\frac{x + 5}{7} \div \frac{4x + 20}{9}$

Verified step by step guidance

1. Rewrite the division of fractions as multiplication by the reciprocal. So, \( \frac{(x+5)}{7} \div \frac{(4x+20)}{9} \) becomes \( \frac{(x+5)}{7} \times \frac{9}{(4x+20)} \).
2. Factor any expressions that can be simplified. Notice that \(4x + 20\) can be factored as \(4(x + 5)\).
3. Substitute the factored form back into the expression: \( \frac{(x+5)}{7} \times \frac{9}{4(x+5)} \).
4. Cancel out the common factor \( (x+5) \) from numerator and denominator, assuming \(x \neq -5\) to avoid division by zero.
5. Multiply the remaining numerators and denominators: \( \frac{1 \times 9}{7 \times 4} \) to get the simplified expression.

Key Concepts

Division of Rational Expressions

Dividing rational expressions involves multiplying the first expression by the reciprocal of the second. Instead of directly dividing, you flip the numerator and denominator of the divisor and then multiply, simplifying the process.

Factoring Polynomials

Factoring polynomials means rewriting them as a product of simpler expressions. For example, 4x + 20 can be factored as 4(x + 5), which helps in simplifying rational expressions by canceling common factors.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form by canceling common factors in the numerator and denominator. This makes the expression easier to work with and understand.