# How do you combine (3x+1)/(x-2) - (4x+1)/(x-3)?

May 27, 2018

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

#### Explanation:

We have
$\frac{3 x + 1}{x - 2} - \frac{4 x + 1}{x - 3} = \frac{\left(3 x + 1\right) \left(x - 3\right) - \left(4 x + 1\right) \left(x - 2\right)}{\left(x - 2\right) \left(x - 3\right)}$
Expanding the numerator:
$\left(3 x + 1\right) \left(x - 3\right) - \left(4 x + 1\right) \left(x - 2\right) = 3 {x}^{2} + x - 9 x - 3 - \left(4 {x}^{2} + x - 8 x - 2\right) =$
$3 {x}^{2} - 8 x - 3 - 4 {x}^{2} + 7 x + 2 = - {x}^{2} - x - 1$

May 27, 2018

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

#### Explanation:

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

color(green)( color(white)("dd")[(3x+1)/(x-2)color(red)(xx1)]color(white)("ddd")-color(white)("dd")[(4x+1)/(x-3)color(red)(xx1)]

$\textcolor{g r e e n}{\left[\frac{3 x + 1}{x - 2} \textcolor{red}{\times \frac{x - 3}{x - 3}}\right] \textcolor{w h i t e}{\text{d}} - \left[\frac{4 x + 1}{x - 3} \textcolor{red}{\times \frac{x - 2}{x - 2}}\right]}$

$\textcolor{g r e e n}{\left[\frac{3 {x}^{2} - 9 x + x - 3}{\left(x - 3\right) \left(x - 2\right)}\right] - \left[\frac{4 {x}^{2} - 8 x + x - 2}{\left(x - 3\right) \left(x - 2\right)}\right]}$

$\textcolor{g r e e n}{\left[\frac{3 {x}^{2} - 8 x - 3}{\left(x - 3\right) \left(x - 2\right)}\right] - \left[\frac{4 {x}^{2} - 7 x - 2}{\left(x - 3\right) \left(x - 2\right)}\right]}$

color(white)("ddddddddd")color(green)( (-x^2-x-1)/((x-3)(x-2) )

color(white)("ddddddddd")color(green)( (-(x^2+x+1))/((x-3)(x-2) )

color(white)("dddddddd")color(green)( -(x^2+x+1)/((x-3)(x-2) )