Simplify ((x+4)/3+1/x)/(1+1/x)

Solution:

Step 1:

Cross multiply the fraction by $3x$.

Step 1.1:

Multiply $\frac{\frac{x + 4}{3} + \frac{1}{x}}{1 + \frac{1}{x}}$ by $\frac{3x}{3x}$ to get $\frac{3x}{3x} \cdot \frac{\frac{x + 4}{3} + \frac{1}{x}}{1 + \frac{1}{x}}$.

Step 1.2:

Combine the terms to obtain $\frac{3x \left( \frac{x + 4}{3} + \frac{1}{x} \right)}{3x \left( 1 + \frac{1}{x} \right)}$.

Step 2:

Distribute $3x$ across the terms in the numerator and denominator as $\frac{3x \cdot \frac{x + 4}{3} + 3x \cdot \frac{1}{x}}{3x \cdot 1 + 3x \cdot \frac{1}{x}}$.

Step 3:

Proceed to cancel out common factors.

Step 3.1:

Eliminate the factor of $3$.

Step 3.1.1:

Extract $3$ from $3x$ to get $\frac{3(x) \cdot \frac{x + 4}{3} + 3x \cdot \frac{1}{x}}{3x \cdot 1 + 3x \cdot \frac{1}{x}}$.

Step 3.1.2:

Cancel out the common factor to simplify to $\frac{x(x + 4) + 3x \cdot \frac{1}{x}}{3x \cdot 1 + 3x \cdot \frac{1}{x}}$.

Step 3.1.3:

Rewrite the expression as $\frac{x(x + 4) + 3}{3x \cdot 1 + 3x \cdot \frac{1}{x}}$.

Step 3.2:

Remove the common factor of $x$.

Step 3.2.1:

Factor $x$ out of $3x$ to obtain $\frac{x(x + 4) + x \cdot 3 \cdot \frac{1}{x}}{3x \cdot 1 + 3x \cdot \frac{1}{x}}$.

Step 3.2.2:

Cancel out the common factor to simplify to $\frac{x(x + 4) + 3}{3x \cdot 1 + 3x \cdot \frac{1}{x}}$.

Step 3.2.3:

Rewrite the expression as $\frac{x(x + 4) + 3}{3x + 3}$.

Step 3.3:

Eliminate the common factor of $x$.

Step 3.3.1:

Factor $x$ out of $3x$ to get $\frac{x(x + 4) + 3}{3x \cdot 1 + x \cdot 3 \cdot \frac{1}{x}}$.

Step 3.3.2:

Cancel out the common factor to simplify to $\frac{x(x + 4) + 3}{3x + 3}$.

Step 3.3.3:

Rewrite the expression as $\frac{x(x + 4) + 3}{3x + 3}$.

Step 4:

Simplify the numerator.

Step 4.1:

Apply the distributive property to get $\frac{x \cdot x + x \cdot 4 + 3}{3x + 3}$.

Step 4.2:

Multiply $x$ by $x$ to obtain $\frac{x^2 + x \cdot 4 + 3}{3x + 3}$.

Step 4.3:

Rearrange to $\frac{x^2 + 4x + 3}{3x + 3}$.

Step 4.4:

Factor $x^2 + 4x + 3$ using the AC method.

Step 4.4.1:

Identify integers with a product of $3$ and a sum of $4$, which are $1$ and $3$.

Step 4.4.2:

Write the factored form as $\frac{(x + 1)(x + 3)}{3x + 3}$.

Step 5:

Simplify the denominator.

Step 5.1:

Factor out $3$ from $3x + 3$.

Step 5.1.1:

Factor $3$ out of $3x$ to get $\frac{(x + 1)(x + 3)}{3(x + 1)}$.

Step 6:

Cancel the common factor of $x + 1$.

Step 6.1:

Cancel the common factor to simplify to $\frac{x + 3}{3}$.

Knowledge Notes:

To simplify the given complex fraction, we use the following knowledge points:

1. Multiplying by a form of one: Multiplying by $\frac{3x}{3x}$ does not change the value of the fraction but allows us to clear the complex fraction by getting rid of the denominators within the numerator and the denominator.

2. Distributive property: This property allows us to multiply a single term by each term within a set of parentheses, which is used to simplify expressions.

3. Cancelling common factors: When the same factor appears in both the numerator and the denominator, it can be cancelled out to simplify the fraction.

4. Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions. In this case, we use the AC method to factor a quadratic expression.

5. AC method: This is a technique used to factor quadratic expressions of the form $ax^2 + bx + c$. It involves finding two numbers that multiply to give $ac$ and add to give $b$.

By applying these principles, we can simplify the original complex fraction to a much simpler form.