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

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(x + 2) (4 x - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x - 2}{x + 2}$ by $\frac{4 x - 1}{4 x - 1}$ .

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

Multiply $\frac{2 x}{4 x - 1}$ by $\frac{x + 2}{x + 2}$ .

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

Reorder the factors of $(x + 2) (4 x - 1)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Expand $(3 x - 2) (4 x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $4$ .

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

Multiply $- 1$ by $3$ .

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

Multiply $4$ by $- 2$ .

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

Multiply $- 2$ by $- 1$ .

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

Subtract $8 x$ from $- 3 x$ .

$\frac{12 x^{2} - 11 x + 2 + 2 x (x + 2)}{(4 x - 1) (x + 2)}$

Apply the distributive property.

$\frac{12 x^{2} - 11 x + 2 + 2 x \cdot x + 2 x \cdot 2}{(4 x - 1) (x + 2)}$

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

Move $x$ .

$\frac{12 x^{2} - 11 x + 2 + 2 (x \cdot x) + 2 x \cdot 2}{(4 x - 1) (x + 2)}$

Multiply $x$ by $x$ .

$\frac{12 x^{2} - 11 x + 2 + 2 x^{2} + 2 x \cdot 2}{(4 x - 1) (x + 2)}$

Multiply $2$ by $2$ .

$\frac{12 x^{2} - 11 x + 2 + 2 x^{2} + 4 x}{(4 x - 1) (x + 2)}$

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

$\frac{14 x^{2} - 11 x + 2 + 4 x}{(4 x - 1) (x + 2)}$

Add $- 11 x$ and $4 x$ .

$\frac{14 x^{2} - 7 x + 2}{(4 x - 1) (x + 2)}$

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

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