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

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

Evaluate

$$\frac{4x-7}{\left(x-1\right)\left(x^{2}-4\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Express \frac{\frac{4x-7}{\left(x-2\right)\left(x-1\right)}}{x+2} as a single fraction.

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

Apply the distributive property by multiplying each term of x-2 by each term of x-1.

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

Combine -x and -2x to get -3x.

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

Apply the distributive property by multiplying each term of x^{2}-3x+2 by each term of x+2.

$$\frac{4x-7}{x^{3}+2x^{2}-3x^{2}-6x+2x+4}$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$\frac{4x-7}{x^{3}-x^{2}-6x+2x+4}$$

Combine -6x and 2x to get -4x.

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

Expand

$$\frac{4x-7}{\left(x-1\right)\left(x^{2}-4\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Express \frac{\frac{4x-7}{\left(x-2\right)\left(x-1\right)}}{x+2} as a single fraction.

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

Apply the distributive property by multiplying each term of x-2 by each term of x-1.

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

Combine -x and -2x to get -3x.

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

Apply the distributive property by multiplying each term of x^{2}-3x+2 by each term of x+2.

$$\frac{4x-7}{x^{3}+2x^{2}-3x^{2}-6x+2x+4}$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$\frac{4x-7}{x^{3}-x^{2}-6x+2x+4}$$

Combine -6x and 2x to get -4x.

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