Solve 2x/x-3+1/2x+3+3x+9/(x-3)(2x+3)

$$\frac { 2 x } { x - 3 } + \frac { 1 } { 2 x + 3 } + \frac { 3 x + 9 } { ( x - 3 ) ( 2 x + 3 ) }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 2x+3 is \left(x-3\right)\left(2x+3\right). Multiply \frac{2x}{x-3} times \frac{2x+3}{2x+3}. Multiply \frac{1}{2x+3} times \frac{x-3}{x-3}.

$$\frac{2x\left(2x+3\right)}{\left(x-3\right)\left(2x+3\right)}+\frac{x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Since \frac{2x\left(2x+3\right)}{\left(x-3\right)\left(2x+3\right)} and \frac{x-3}{\left(x-3\right)\left(2x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x\left(2x+3\right)+x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Do the multiplications in 2x\left(2x+3\right)+x-3.

$$\frac{4x^{2}+6x+x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Combine like terms in 4x^{2}+6x+x-3.

$$\frac{4x^{2}+7x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Since \frac{4x^{2}+7x-3}{\left(x-3\right)\left(2x+3\right)} and \frac{3x+9}{\left(x-3\right)\left(2x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x^{2}+7x-3+3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Combine like terms in 4x^{2}+7x-3+3x+9.

$$\frac{4x^{2}+10x+6}{\left(x-3\right)\left(2x+3\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}+10x+6}{\left(x-3\right)\left(2x+3\right)}.

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

Cancel out 2x+3 in both numerator and denominator.

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

Use the distributive property to multiply 2 by x+1.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 2x+3 is \left(x-3\right)\left(2x+3\right). Multiply \frac{2x}{x-3} times \frac{2x+3}{2x+3}. Multiply \frac{1}{2x+3} times \frac{x-3}{x-3}.

$$\frac{2x\left(2x+3\right)}{\left(x-3\right)\left(2x+3\right)}+\frac{x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Since \frac{2x\left(2x+3\right)}{\left(x-3\right)\left(2x+3\right)} and \frac{x-3}{\left(x-3\right)\left(2x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x\left(2x+3\right)+x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Do the multiplications in 2x\left(2x+3\right)+x-3.

$$\frac{4x^{2}+6x+x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Combine like terms in 4x^{2}+6x+x-3.

$$\frac{4x^{2}+7x-3}{\left(x-3\right)\left(2x+3\right)}+\frac{3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Since \frac{4x^{2}+7x-3}{\left(x-3\right)\left(2x+3\right)} and \frac{3x+9}{\left(x-3\right)\left(2x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x^{2}+7x-3+3x+9}{\left(x-3\right)\left(2x+3\right)}$$

Combine like terms in 4x^{2}+7x-3+3x+9.

$$\frac{4x^{2}+10x+6}{\left(x-3\right)\left(2x+3\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}+10x+6}{\left(x-3\right)\left(2x+3\right)}.

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

Cancel out 2x+3 in both numerator and denominator.

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

Use the distributive property to multiply 2 by x+1.

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