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

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

Evaluate

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

Short Solution Steps

Convert 1 to fraction \frac{2}{2}.

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

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

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

Add 2 and 1 to get 3.

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

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

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

Since \frac{3\left(x+1\right)}{2\left(x+1\right)} and \frac{2}{2\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x+1\right)-2.

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

Combine like terms in 3x+3-2.

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

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

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

Since \frac{\left(3x+1\right)\left(x+2\right)}{2\left(x+1\right)\left(x+2\right)} and \frac{2\left(x+1\right)}{2\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x+1\right)\left(x+2\right)-2\left(x+1\right).

$$\frac{3}{2}\times \frac{3x^{2}+6x+x+2-2x-2}{2\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{3}{2}\times \frac{3x^{2}+5x}{2\left(x+1\right)\left(x+2\right)}$$

Multiply \frac{3}{2} times \frac{3x^{2}+5x}{2\left(x+1\right)\left(x+2\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(3x^{2}+5x\right)}{2\times 2\left(x+1\right)\left(x+2\right)}$$

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 3 by 3x^{2}+5x.

$$\frac{9x^{2}+15x}{4\left(x+1\right)\left(x+2\right)}$$

Use the distributive property to multiply 4 by x+1.

$$\frac{9x^{2}+15x}{\left(4x+4\right)\left(x+2\right)}$$

Apply the distributive property by multiplying each term of 4x+4 by each term of x+2.

$$\frac{9x^{2}+15x}{4x^{2}+8x+4x+8}$$

Combine 8x and 4x to get 12x.

$$\frac{9x^{2}+15x}{4x^{2}+12x+8}$$

Expand

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

Short Solution Steps

Convert 1 to fraction \frac{2}{2}.

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

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

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

Add 2 and 1 to get 3.

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

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

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

Since \frac{3\left(x+1\right)}{2\left(x+1\right)} and \frac{2}{2\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x+1\right)-2.

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

Combine like terms in 3x+3-2.

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

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

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

Since \frac{\left(3x+1\right)\left(x+2\right)}{2\left(x+1\right)\left(x+2\right)} and \frac{2\left(x+1\right)}{2\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x+1\right)\left(x+2\right)-2\left(x+1\right).

$$\frac{3}{2}\times \frac{3x^{2}+6x+x+2-2x-2}{2\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{3}{2}\times \frac{3x^{2}+5x}{2\left(x+1\right)\left(x+2\right)}$$

Multiply \frac{3}{2} times \frac{3x^{2}+5x}{2\left(x+1\right)\left(x+2\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(3x^{2}+5x\right)}{2\times 2\left(x+1\right)\left(x+2\right)}$$

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 3 by 3x^{2}+5x.

$$\frac{9x^{2}+15x}{4\left(x+1\right)\left(x+2\right)}$$

Use the distributive property to multiply 4 by x+1.

$$\frac{9x^{2}+15x}{\left(4x+4\right)\left(x+2\right)}$$

Apply the distributive property by multiplying each term of 4x+4 by each term of x+2.

$$\frac{9x^{2}+15x}{4x^{2}+8x+4x+8}$$

Combine 8x and 4x to get 12x.

$$\frac{9x^{2}+15x}{4x^{2}+12x+8}$$