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

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

Step 1

To write $\frac{3 x - 1}{2 x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 2

To write $\frac{3 x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

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

Step 3

Write each expression with a common denominator of $2 x (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x - 1}{2 x}$ by $\frac{x - 1}{x - 1}$ .

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

Multiply $\frac{3 x}{x - 1}$ by $\frac{2 x}{2 x}$ .

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

Reorder the factors of $(x - 1) (2 x)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Expand $(3 x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $3$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- 3 x$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $2$ .

$\frac{3 x^{2} - 4 x + 1 + 6 x^{2}}{2 x (x - 1)}$

Add $3 x^{2}$ and $6 x^{2}$ .

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

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

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