In Exercises 15–32, multiply or divide as indicated. (x+5)/7 ÷ (4x+20)/9

Verified step by step guidance

1. Identify the operation: We need to divide two rational expressions, \( \frac{x+5}{7} \) and \( \frac{4x+20}{9} \).
2. Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, rewrite the division as a multiplication: \( \frac{x+5}{7} \times \frac{9}{4x+20} \).
3. Simplify the expression \( \frac{4x+20}{9} \) by factoring the numerator. Notice that \( 4x+20 = 4(x+5) \).
4. Substitute the factored form into the expression: \( \frac{x+5}{7} \times \frac{9}{4(x+5)} \).
5. Cancel the common factor \( x+5 \) from the numerator and the denominator, then multiply the remaining fractions.

Key Concepts

Fraction Division

Dividing fractions involves multiplying by the reciprocal of the divisor. In this case, to divide (x+5)/7 by (4x+20)/9, you first rewrite the division as multiplication by the reciprocal: (x+5)/7 * (9/(4x+20)). This process simplifies the operation and allows for easier manipulation of the fractions.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the expression (4x+20), you can factor out the common term, resulting in 4(x+5). This simplification is crucial for reducing fractions and making calculations more manageable.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling out common factors in the numerator and denominator. After rewriting the division as multiplication and factoring, you can cancel out the (x+5) terms, leading to a more straightforward expression that is easier to evaluate or further manipulate.