حل {l}{(m+n)x+3m-2n<0}{x>-1/2}{(m+2n)x+n-2m>0}

$$\left. \begin{array} { l } { ( m + n ) x + 3 m - 2 n < 0 } \\ { x > - \frac { 1 } { 2 } } \\ { ( m + 2 n ) x + n - 2 m > 0 } \end{array} \right.$$

برای x حل کنید

$$\left\{\begin{matrix}x\in \left(-\frac{1}{2},-\frac{3m-2n}{m+n}\right)\text{, }&\left(m>-n\text{ and }m\leq 0\right)\text{ or }\left(m\leq \frac{\sqrt{5}|n|-n}{2}\text{ and }m\geq \frac{-\sqrt{5}|n|-n}{2}\text{ and }m<-2n\text{ and }m>-n\text{ and }|n|>|m|\right)\\x\in \left(-\frac{1}{2},-\frac{n-2m}{m+2n}\right)\text{, }&\left(m<-2n\text{ and }m<-|n|\right)\text{ or }\left(m<0\text{ and }m<-2n\text{ and }m>\frac{\sqrt{5}|n|-n}{2}\text{ and }m>-n\text{ and }|n|>-m\right)\text{ or }\left(m<0\text{ and }m<-2n\text{ and }m<\frac{-\sqrt{5}|n|-n}{2}\text{ and }m>-n\text{ and }|n|>-m\right)\\x\in \left(-\frac{n-2m}{m+2n},-\frac{3m-2n}{m+n}\right)\text{, }&m>0\text{ and }m>-2n\text{ and }m>\frac{-\sqrt{5}|n|-n}{2}\text{ and }m<\frac{\sqrt{5}|n|-n}{2}\text{ and }m>-n\text{ and }|n|>m\\x>-\frac{1}{2}\text{, }&\left(m>-2n\text{ and }m<-|n|\right)\text{ or }\left(n>0\text{ and }m=-2n\right)\text{ or }\left(m=-n\text{ and }n>0\right)\\x\in \left(-\frac{3m-2n}{m+n},-\frac{n-2m}{m+2n}\right)\text{, }&\left(m=n\text{ and }n<0\right)\text{ or }\left(m<\frac{-\sqrt{5}|n|-n}{2}\text{ and }m<-2n\text{ and }m<-n\text{ and }m\geq n\right)\text{ or }\left(m>\frac{\sqrt{5}|n|-n}{2}\text{ and }m<-2n\text{ and }m<-n\text{ and }m\geq n\right)\\x>-\frac{3m-2n}{m+n}\text{, }&\left(m\leq \frac{-\sqrt{5}|n|-n}{2}\text{ and }m<-n\text{ and }m>-2n\text{ and }m\geq n\right)\text{ or }\left(m\geq \frac{\sqrt{5}|n|-n}{2}\text{ and }m<-n\text{ and }m>-2n\text{ and }m\geq n\right)\\x>-\frac{n-2m}{m+2n}\text{, }&m>\frac{-\sqrt{5}|n|-n}{2}\text{ and }m<\frac{\sqrt{5}|n|-n}{2}\text{ and }m<-n\text{ and }m>-2n\text{ and }m\geq n\end{matrix}\right.$$

برای n حل کنید

$$\left\{\begin{matrix}n\in \left(-\frac{m\left(x-2\right)}{2x+1},-\frac{m\left(x+3\right)}{x-2}\right)\text{, }&x>2\text{ and }m<0\\n>0\text{, }&m<0\text{ and }x=2\\n>-\frac{m\left(x-2\right)}{2x+1}\text{, }&\left(m>0\text{ and }x<\frac{5\sqrt{5}-11}{2}\text{ and }x>-\frac{1}{2}\right)\text{ or }\left(m<0\text{ and }x>\frac{5\sqrt{5}-11}{2}\text{ and }x<2\right)\\n>-\frac{m\left(x+3\right)}{x-2}\text{, }&\left(x\geq \frac{5\sqrt{5}-11}{2}\text{ and }m\geq 0\text{ and }x<2\right)\text{ or }x=\frac{5\sqrt{5}-11}{2}\text{ or }\left(m=0\text{ and }x<2\text{ and }x>-\frac{1}{2}\right)\text{ or }\left(m\leq 0\text{ and }x\leq \frac{5\sqrt{5}-11}{2}\text{ and }x>-\frac{1}{2}\right)\end{matrix}\right.$$