Solve x-7/x^2+4x-5-x-9/x^2+3x-10

$$\frac { x - 7 } { x ^ { 2 } + 4 x - 5 } - \frac { x - 9 } { x ^ { 2 } + 3 x - 10 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+4x-5. Factor x^{2}+3x-10.

$$\frac{x-7}{\left(x-1\right)\left(x+5\right)}-\frac{x-9}{\left(x-2\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+5\right) and \left(x-2\right)\left(x+5\right) is \left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{x-7}{\left(x-1\right)\left(x+5\right)} times \frac{x-2}{x-2}. Multiply \frac{x-9}{\left(x-2\right)\left(x+5\right)} times \frac{x-1}{x-1}.

$$\frac{\left(x-7\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}-\frac{\left(x-9\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Since \frac{\left(x-7\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{\left(x-9\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-7\right)\left(x-2\right)-\left(x-9\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Do the multiplications in \left(x-7\right)\left(x-2\right)-\left(x-9\right)\left(x-1\right).

$$\frac{x^{2}-2x-7x+14-x^{2}+x+9x-9}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Combine like terms in x^{2}-2x-7x+14-x^{2}+x+9x-9.

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

Cancel out x+5 in both numerator and denominator.

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

Expand \left(x-2\right)\left(x-1\right).

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

Expand

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

Short Solution Steps

Factor x^{2}+4x-5. Factor x^{2}+3x-10.

$$\frac{x-7}{\left(x-1\right)\left(x+5\right)}-\frac{x-9}{\left(x-2\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+5\right) and \left(x-2\right)\left(x+5\right) is \left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{x-7}{\left(x-1\right)\left(x+5\right)} times \frac{x-2}{x-2}. Multiply \frac{x-9}{\left(x-2\right)\left(x+5\right)} times \frac{x-1}{x-1}.

$$\frac{\left(x-7\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}-\frac{\left(x-9\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Since \frac{\left(x-7\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{\left(x-9\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-7\right)\left(x-2\right)-\left(x-9\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Do the multiplications in \left(x-7\right)\left(x-2\right)-\left(x-9\right)\left(x-1\right).

$$\frac{x^{2}-2x-7x+14-x^{2}+x+9x-9}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Combine like terms in x^{2}-2x-7x+14-x^{2}+x+9x-9.

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

Cancel out x+5 in both numerator and denominator.

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

Expand \left(x-2\right)\left(x-1\right).

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