Simplify (x+2)/(x-4)+2/x+(5x)/(3x-1)

$\frac{x + 2}{x - 4} + \frac{2}{x} + \frac{5 x}{3 x - 1}$

Step 1

To write $\frac{x + 2}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x + 2}{x - 4} \cdot \frac{x}{x} + \frac{2}{x} + \frac{5 x}{3 x - 1}$

Step 2

To write $\frac{2}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{x + 2}{x - 4} \cdot \frac{x}{x} + \frac{2}{x} \cdot \frac{x - 4}{x - 4} + \frac{5 x}{3 x - 1}$

Step 3

Write each expression with a common denominator of $(x - 4) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 2}{x - 4}$ by $\frac{x}{x}$ .

$\frac{(x + 2) x}{(x - 4) x} + \frac{2}{x} \cdot \frac{x - 4}{x - 4} + \frac{5 x}{3 x - 1}$

Multiply $\frac{2}{x}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(x + 2) x}{(x - 4) x} + \frac{2 (x - 4)}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Reorder the factors of $(x - 4) x$ .

$\frac{(x + 2) x}{x (x - 4)} + \frac{2 (x - 4)}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 2) x + 2 (x - 4)}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot x + 2 x + 2 (x - 4)}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Multiply $x$ by $x$ .

$\frac{x^{2} + 2 x + 2 (x - 4)}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Apply the distributive property.

$\frac{x^{2} + 2 x + 2 x + 2 \cdot - 4}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Multiply $2$ by $- 4$ .

$\frac{x^{2} + 2 x + 2 x - 8}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Add $2 x$ and $2 x$ .

$\frac{x^{2} + 4 x - 8}{x (x - 4)} + \frac{5 x}{3 x - 1}$

Step 6

To write $\frac{x^{2} + 4 x - 8}{x (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{3 x - 1}{3 x - 1}$ .

$\frac{x^{2} + 4 x - 8}{x (x - 4)} \cdot \frac{3 x - 1}{3 x - 1} + \frac{5 x}{3 x - 1}$

Step 7

To write $\frac{5 x}{3 x - 1}$ as a fraction with a common denominator, multiply by $\frac{x (x - 4)}{x (x - 4)}$ .

$\frac{x^{2} + 4 x - 8}{x (x - 4)} \cdot \frac{3 x - 1}{3 x - 1} + \frac{5 x}{3 x - 1} \cdot \frac{x (x - 4)}{x (x - 4)}$

Step 8

Write each expression with a common denominator of $x (x - 4) (3 x - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x^{2} + 4 x - 8}{x (x - 4)}$ by $\frac{3 x - 1}{3 x - 1}$ .

$\frac{(x^{2} + 4 x - 8) (3 x - 1)}{x (x - 4) (3 x - 1)} + \frac{5 x}{3 x - 1} \cdot \frac{x (x - 4)}{x (x - 4)}$

Multiply $\frac{5 x}{3 x - 1}$ by $\frac{x (x - 4)}{x (x - 4)}$ .

$\frac{(x^{2} + 4 x - 8) (3 x - 1)}{x (x - 4) (3 x - 1)} + \frac{5 x (x (x - 4))}{(3 x - 1) (x (x - 4))}$

Reorder the factors of $x (x - 4) (3 x - 1)$ .

$\frac{(x^{2} + 4 x - 8) (3 x - 1)}{x (3 x - 1) (x - 4)} + \frac{5 x (x (x - 4))}{(3 x - 1) (x (x - 4))}$

Reorder the factors of $(3 x - 1) (x (x - 4))$ .

$\frac{(x^{2} + 4 x - 8) (3 x - 1)}{x (3 x - 1) (x - 4)} + \frac{5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Step 9

Combine the numerators over the common denominator.

$\frac{(x^{2} + 4 x - 8) (3 x - 1) + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Step 10

Simplify the numerator.

Expand $(x^{2} + 4 x - 8) (3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (3 x) + x^{2} \cdot - 1 + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{3 x^{2} x + x^{2} \cdot - 1 + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 (x \cdot x^{2}) + x^{2} \cdot - 1 + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$\frac{3 (x^{1} x^{2}) + x^{2} \cdot - 1 + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 x^{1 + 2} + x^{2} \cdot - 1 + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Add $1$ and $2$ .

$\frac{3 x^{3} + x^{2} \cdot - 1 + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Move $- 1$ to the left of $x^{2}$ .

$\frac{3 x^{3} - 1 \cdot x^{2} + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{3 x^{3} - x^{2} + 4 x (3 x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Rewrite using the commutative property of multiplication.

$\frac{3 x^{3} - x^{2} + 4 \cdot 3 x \cdot x + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 x^{3} - x^{2} + 4 \cdot 3 (x \cdot x) + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{3 x^{3} - x^{2} + 4 \cdot 3 x^{2} + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $4$ by $3$ .

$\frac{3 x^{3} - x^{2} + 12 x^{2} + 4 x \cdot - 1 - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $- 1$ by $4$ .

$\frac{3 x^{3} - x^{2} + 12 x^{2} - 4 x - 8 (3 x) - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $3$ by $- 8$ .

$\frac{3 x^{3} - x^{2} + 12 x^{2} - 4 x - 24 x - 8 \cdot - 1 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $- 8$ by $- 1$ .

$\frac{3 x^{3} - x^{2} + 12 x^{2} - 4 x - 24 x + 8 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Add $- x^{2}$ and $12 x^{2}$ .

$\frac{3 x^{3} + 11 x^{2} - 4 x - 24 x + 8 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Subtract $24 x$ from $- 4 x$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 x (x (x - 4))}{x (3 x - 1) (x - 4)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 (x \cdot x) (x - 4)}{x (3 x - 1) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 x^{2} (x - 4)}{x (3 x - 1) (x - 4)}$

Apply the distributive property.

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 x^{2} x + 5 x^{2} \cdot - 4}{x (3 x - 1) (x - 4)}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 (x \cdot x^{2}) + 5 x^{2} \cdot - 4}{x (3 x - 1) (x - 4)}$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 (x^{1} x^{2}) + 5 x^{2} \cdot - 4}{x (3 x - 1) (x - 4)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 x^{1 + 2} + 5 x^{2} \cdot - 4}{x (3 x - 1) (x - 4)}$

Add $1$ and $2$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 x^{3} + 5 x^{2} \cdot - 4}{x (3 x - 1) (x - 4)}$

Multiply $- 4$ by $5$ .

$\frac{3 x^{3} + 11 x^{2} - 28 x + 8 + 5 x^{3} - 20 x^{2}}{x (3 x - 1) (x - 4)}$

Add $3 x^{3}$ and $5 x^{3}$ .

$\frac{8 x^{3} + 11 x^{2} - 28 x + 8 - 20 x^{2}}{x (3 x - 1) (x - 4)}$

Subtract $20 x^{2}$ from $11 x^{2}$ .

$\frac{8 x^{3} - 9 x^{2} - 28 x + 8}{x (3 x - 1) (x - 4)}$

Simplify (x+2)/(x-4)+2/x+(5x)/(3x-1)

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