# How do you solve 2/(x+1) + 5/(x-2)=-2?

Apr 17, 2018

$x = - 3 , \frac{1}{2}$

#### Explanation:

1. Find the least common denominator of the left side. Since we have expressions containing variables, we just multiply both denominators:
$\setminus \textcolor{red}{\left(x + 1\right) \left(x - 2\right)}$
by the way, if you FOIL this it becomes $\setminus \textcolor{\mathmr{and} \chi d}{{x}^{2} - x - 2}$

2. Multiply everything by that least common denominator.
\color(red)((x+1)(x-2))[2/(x+1)+5/(x-2)]=\color(red)((x+1)(x-2)) (-2)

3. Apply the distributive property:

4. (\color(red)((x+1)(x-2))(2))/(x+1)+(\color(red)((x+1)(x-2)) (5))/(x-2)=(\color(orchid)(x^2-x-2)) (-2)
5. ((x+1)(x-2)(2))/(x+1)+((x+1)(x-2) (5))/(x-2)=\color(orchid)(-2x^2+2x+4)

6. Cancel out like terms and simplify:

7. ((\cancel(x+1))(x-2)(2))/\cancel(x+1)+((x+1)\cancel((x-2)) (5))/\cancel(x-2)=-2x^2+2x+4
8. $\left(x - 2\right) \left(2\right) + \left(x + 1\right) \left(5\right) = - 2 {x}^{2} + 2 x + 4$

9. Re-apply distributive property:

10. $\setminus \textcolor{\to m a \to}{\left(x - 2\right) \left(2\right)} + \setminus \textcolor{s e a g r e e n}{\left(x + 1\right) \left(5\right)} = - 2 {x}^{2} + 2 x + 4$
11. $\setminus \textcolor{\to m a \to}{2 x - 4} + \setminus \textcolor{s e a g r e e n}{5 x + 5} = - 2 {x}^{2} + 2 x + 4$

12. Identify like terms and combine:

13. $\setminus \textcolor{h o t \pi n k}{2 x} \setminus \textcolor{s t e e l b l u e}{- 4} + \setminus \textcolor{h o t \pi n k}{5 x} + \setminus \textcolor{s t e e l b l u e}{5} = - 2 {x}^{2} + \setminus \textcolor{h o t \pi n k}{2 x} + \setminus \textcolor{s t e e l b l u e}{4}$
14. $\setminus \textcolor{h o t \pi n k}{7 x} + \setminus \textcolor{s t e e l b l u e}{1} = - 2 {x}^{2} + \setminus \textcolor{h o t \pi n k}{2 x} + \setminus \textcolor{s t e e l b l u e}{4}$
15. $2 {x}^{2} + \setminus \textcolor{h o t \pi n k}{7 x - 2 x} + \setminus \textcolor{s t e e l b l u e}{1 - 4} = 0$
16. $2 {x}^{2} + \setminus \textcolor{h o t \pi n k}{5 x} + \left(\setminus \textcolor{s t e e l b l u e}{- 3}\right) = 0$
17. $2 {x}^{2} + 5 x - 3 = 0$ And now you have to solve this polynomial.
18. Solve the polynomial from the last step (see here for details).