rational expression

[Guest]

I am having trouble doing this problem:

Solve the equation involving rational expressions. Verify your solutions.

x-3/x+2 + 2/3=0

(sorry if I didn't write this correctly).

 

[soroban]

Hello, angelasloan7038!

Solve the equation; verify your solution.

\(\displaystyle \L\;\;\;\frac{x\,-\,3}{x\,+\,2}\,+\,\frac{2}{3}\:=\:0\)

We have: \(\displaystyle \L\;\frac{x\,-\,3}{x\,+\,2}\:=\:-\frac{2}{3}\)

Cross-multiply: \(\displaystyle \;3(x\,-\,3)\:=\:-2(x\,+\,2)\)

. . . \(\displaystyle 3x\,-\,9\:=\:-2x\,-\,4\;\;\Rightarrow\;\;5x\,=\,5\;\;\Rightarrow\;\;x\,=\,1\)

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Check: .Let \(\displaystyle x = 1\)

Does \(\displaystyle \L\;\frac{1\,-\,3}{1\,+\,2}\,+\,\frac{2}{3}\;=\;0\) ?

 

[jsbeckton]

\(\displaystyle \begin{array}{l}
\frac{{x - 3}}{{x + 2}} + \frac{2}{3} = 0 \\
\frac{{x - 3}}{{x + 2}} = - \frac{2}{3} \\
3(x - 3) = - 2(x + 2) \\
3x - 9 = - 2x - 4 \\
5x - 9 = - 4 \\
5x = 5 \\
x = 1 \\
{\rm you can verify that this is correct by subbing for x} \\
{\rm in the original equation:} \\
\frac{{(1) - 3}}{{(1) + 2}} + \frac{2}{3} = 0 \\
\frac{{ - 2}}{3} + \frac{2}{3} = 0 \\
\end{array}\)