Solve x+5/2x-2-2x+7/x^2+x-2+3/4

$$\frac { x + 5 } { 2 x - 2 } - \frac { 2 x + 7 } { x ^ { 2 } + x - 2 } + \frac { 3 } { 4 }$$

Evaluate

$$\frac{5x+14}{4\left(x+2\right)}$$

Short Solution Steps

Factor 2x-2. Factor x^{2}+x-2.

$$\frac{x+5}{2\left(x-1\right)}-\frac{2x+7}{\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-1\right) and \left(x-1\right)\left(x+2\right) is 2\left(x-1\right)\left(x+2\right). Multiply \frac{x+5}{2\left(x-1\right)} times \frac{x+2}{x+2}. Multiply \frac{2x+7}{\left(x-1\right)\left(x+2\right)} times \frac{2}{2}.

$$\frac{\left(x+5\right)\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)}-\frac{2\left(2x+7\right)}{2\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

Since \frac{\left(x+5\right)\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)} and \frac{2\left(2x+7\right)}{2\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+5\right)\left(x+2\right)-2\left(2x+7\right)}{2\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

Do the multiplications in \left(x+5\right)\left(x+2\right)-2\left(2x+7\right).

$$\frac{x^{2}+2x+5x+10-4x-14}{2\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

Combine like terms in x^{2}+2x+5x+10-4x-14.

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

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{2\left(x-1\right)\left(x+2\right)}.

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

Cancel out x-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x+2\right) and 4 is 4\left(x+2\right). Multiply \frac{x+4}{2\left(x+2\right)} times \frac{2}{2}. Multiply \frac{3}{4} times \frac{x+2}{x+2}.

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

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

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

Do the multiplications in 2\left(x+4\right)+3\left(x+2\right).

$$\frac{2x+8+3x+6}{4\left(x+2\right)}$$

Combine like terms in 2x+8+3x+6.

$$\frac{5x+14}{4\left(x+2\right)}$$

Expand 4\left(x+2\right).

$$\frac{5x+14}{4x+8}$$

Expand

$$\frac{5x+14}{4\left(x+2\right)}$$

Short Solution Steps

Factor 2x-2. Factor x^{2}+x-2.

$$\frac{x+5}{2\left(x-1\right)}-\frac{2x+7}{\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-1\right) and \left(x-1\right)\left(x+2\right) is 2\left(x-1\right)\left(x+2\right). Multiply \frac{x+5}{2\left(x-1\right)} times \frac{x+2}{x+2}. Multiply \frac{2x+7}{\left(x-1\right)\left(x+2\right)} times \frac{2}{2}.

$$\frac{\left(x+5\right)\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)}-\frac{2\left(2x+7\right)}{2\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

Since \frac{\left(x+5\right)\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)} and \frac{2\left(2x+7\right)}{2\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+5\right)\left(x+2\right)-2\left(2x+7\right)}{2\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

Do the multiplications in \left(x+5\right)\left(x+2\right)-2\left(2x+7\right).

$$\frac{x^{2}+2x+5x+10-4x-14}{2\left(x-1\right)\left(x+2\right)}+\frac{3}{4}$$

Combine like terms in x^{2}+2x+5x+10-4x-14.

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

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{2\left(x-1\right)\left(x+2\right)}.

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

Cancel out x-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x+2\right) and 4 is 4\left(x+2\right). Multiply \frac{x+4}{2\left(x+2\right)} times \frac{2}{2}. Multiply \frac{3}{4} times \frac{x+2}{x+2}.

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

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

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

Do the multiplications in 2\left(x+4\right)+3\left(x+2\right).

$$\frac{2x+8+3x+6}{4\left(x+2\right)}$$

Combine like terms in 2x+8+3x+6.

$$\frac{5x+14}{4\left(x+2\right)}$$

Expand 4\left(x+2\right).

$$\frac{5x+14}{4x+8}$$