Simplify (2x)/(5x+4)+(6x)/(2x+3)

$\frac{2 x}{5 x + 4} + \frac{6 x}{2 x + 3}$

Step 1

To write $\frac{2 x}{5 x + 4}$ as a fraction with a common denominator, multiply by $\frac{2 x + 3}{2 x + 3}$ .

$\frac{2 x}{5 x + 4} \cdot \frac{2 x + 3}{2 x + 3} + \frac{6 x}{2 x + 3}$

Step 2

To write $\frac{6 x}{2 x + 3}$ as a fraction with a common denominator, multiply by $\frac{5 x + 4}{5 x + 4}$ .

$\frac{2 x}{5 x + 4} \cdot \frac{2 x + 3}{2 x + 3} + \frac{6 x}{2 x + 3} \cdot \frac{5 x + 4}{5 x + 4}$

Step 3

Write each expression with a common denominator of $(5 x + 4) (2 x + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 x}{5 x + 4}$ by $\frac{2 x + 3}{2 x + 3}$ .

$\frac{2 x (2 x + 3)}{(5 x + 4) (2 x + 3)} + \frac{6 x}{2 x + 3} \cdot \frac{5 x + 4}{5 x + 4}$

Multiply $\frac{6 x}{2 x + 3}$ by $\frac{5 x + 4}{5 x + 4}$ .

$\frac{2 x (2 x + 3)}{(5 x + 4) (2 x + 3)} + \frac{6 x (5 x + 4)}{(2 x + 3) (5 x + 4)}$

Reorder the factors of $(5 x + 4) (2 x + 3)$ .

$\frac{2 x (2 x + 3)}{(2 x + 3) (5 x + 4)} + \frac{6 x (5 x + 4)}{(2 x + 3) (5 x + 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 x (2 x + 3) + 6 x (5 x + 4)}{(2 x + 3) (5 x + 4)}$

Step 5

Simplify the numerator.

Factor $2 x$ out of $2 x (2 x + 3) + 6 x (5 x + 4)$ .

Factor $2 x$ out of $6 x (5 x + 4)$ .

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

Factor $2 x$ out of $2 x (2 x + 3) + 2 x (3 (5 x + 4))$ .

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

Apply the distributive property.

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

Multiply $5$ by $3$ .

$\frac{2 x (2 x + 3 + 15 x + 3 \cdot 4)}{(2 x + 3) (5 x + 4)}$

Multiply $3$ by $4$ .

$\frac{2 x (2 x + 3 + 15 x + 12)}{(2 x + 3) (5 x + 4)}$

Add $2 x$ and $15 x$ .

$\frac{2 x (17 x + 3 + 12)}{(2 x + 3) (5 x + 4)}$

Add $3$ and $12$ .

$\frac{2 x (17 x + 15)}{(2 x + 3) (5 x + 4)}$