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

$\frac{\frac{x + 4}{3} + \frac{1}{x}}{1 + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $3 x$ .

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

$\frac{3 x}{3 x} \cdot \frac{\frac{x + 4}{3} + \frac{1}{x}}{1 + \frac{1}{x}}$

Combine.

$\frac{3 x (\frac{x + 4}{3} + \frac{1}{x})}{3 x (1 + \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{3 x \frac{x + 4}{3} + 3 x \frac{1}{x}}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{3 (x) \frac{x + 4}{3} + 3 x \frac{1}{x}}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Cancel the common factor.

$\frac{3 x \frac{x + 4}{3} + 3 x \frac{1}{x}}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Rewrite the expression.

$\frac{x (x + 4) + 3 x \frac{1}{x}}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $3 x$ .

$\frac{x (x + 4) + x \cdot 3 \frac{1}{x}}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Cancel the common factor.

$\frac{x (x + 4) + x \cdot 3 \frac{1}{x}}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Rewrite the expression.

$\frac{x (x + 4) + 3}{3 x \cdot 1 + 3 x \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $3 x$ .

$\frac{x (x + 4) + 3}{3 x \cdot 1 + x \cdot 3 \frac{1}{x}}$

Cancel the common factor.

$\frac{x (x + 4) + 3}{3 x \cdot 1 + x \cdot 3 \frac{1}{x}}$

Rewrite the expression.

$\frac{x (x + 4) + 3}{3 x \cdot 1 + 3}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot x + x \cdot 4 + 3}{3 x \cdot 1 + 3}$

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 4 + 3}{3 x \cdot 1 + 3}$

Move $4$ to the left of $x$ .

$\frac{x^{2} + 4 \cdot x + 3}{3 x \cdot 1 + 3}$

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

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

$\frac{(x + 1) (x + 3)}{3 x \cdot 1 + 3}$

Step 5

Simplify the denominator.

Factor $3$ out of $3 x \cdot 1 + 3$ .

Factor $3$ out of $3 x \cdot 1$ .

$\frac{(x + 1) (x + 3)}{3 (x \cdot 1) + 3}$

Factor $3$ out of $3$ .

$\frac{(x + 1) (x + 3)}{3 (x \cdot 1) + 3 (1)}$

Factor $3$ out of $3 (x \cdot 1) + 3 (1)$ .

$\frac{(x + 1) (x + 3)}{3 (x \cdot 1 + 1)}$

Multiply $x$ by $1$ .

$\frac{(x + 1) (x + 3)}{3 (x + 1)}$

Step 6

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{(x + 1) (x + 3)}{3 (x + 1)}$

Rewrite the expression.

$\frac{x + 3}{3}$

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

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