Simplify (9/(x+3)-3/(x+7))/((x+9)/(x+3))

Solution:

Step 1

Invert the denominator and multiply it by the numerator: $\left(\frac{9}{x + 3} - \frac{3}{x + 7}\right) \cdot \frac{x + 3}{x + 9}$

Step 2

Create a common denominator for $\frac{9}{x + 3}$ by multiplying by $\frac{x + 7}{x + 7}$: $\left(\frac{9 \cdot (x + 7)}{(x + 3)(x + 7)} - \frac{3}{x + 7}\right) \cdot \frac{x + 3}{x + 9}$

Step 3

Create a common denominator for $-\frac{3}{x + 7}$ by multiplying by $\frac{x + 3}{x + 3}$: $\left(\frac{9 \cdot (x + 7)}{(x + 3)(x + 7)} - \frac{3 \cdot (x + 3)}{(x + 3)(x + 7)}\right) \cdot \frac{x + 3}{x + 9}$

Step 4

Combine the fractions over a common denominator: $\frac{9(x + 7) - 3(x + 3)}{(x + 3)(x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 5

Simplify the numerator by distributing and combining like terms:

• Distribute $9$ into $(x + 7)$ and $-3$ into $(x + 3)$: $\frac{9x + 63 - 3x - 9}{(x + 3)(x + 7)} \cdot \frac{x + 3}{x + 9}$
• Combine like terms: $\frac{6x + 54}{(x + 3)(x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 6

Factor out common terms in the numerator: $\frac{6(x + 9)}{(x + 3)(x + 7)} \cdot \frac{x + 3}{x + 9}$

Step 7

Cancel out the common terms in the numerator and denominator:

• Cancel $(x + 9)$: $\frac{6}{(x + 3)(x + 7)} \cdot \frac{x + 3}{1}$
• Cancel $(x + 3)$: $\frac{6}{x + 7}$

The simplified expression is $\frac{6}{x + 7}$.

Knowledge Notes:

To simplify complex fractions, one can follow these steps:

1. Common Denominator: When combining fractions, it's essential to have a common denominator. This can be achieved by multiplying each fraction by an appropriate form of 1, which does not change the value of the fraction but allows the fractions to be combined.

2. Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions and is essential when simplifying the numerator and denominator of a fraction.

3. Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. For example, $ax + bx = (a + b)x$.

4. Factoring: Factoring is the process of breaking down an expression into products of other expressions or numbers. For example, $6(x + 9)$ is factored because it represents $6 \cdot x + 6 \cdot 9$.

5. Cancellation: If a term appears in both the numerator and the denominator of a fraction, it can be cancelled out, as long as it is not equal to zero. This simplifies the fraction.

6. Multiplying by the Reciprocal: To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.

7. Simplification: The goal is to rewrite the expression in the simplest form possible, which often involves combining the above steps to reduce the expression to the lowest terms.