Solve {l}{-40(50-x)leq340}{+a0(50-x)leq29}

$$\left. \begin{array} { l } { - 40 ( 50 - x ) \leq 340 } \\ { + a 0 ( 50 - x ) \leq 29 } \end{array} \right.$$

Solve for x

$$\left\{\begin{matrix}x\leq \frac{1689}{40}\text{, }&a_{0}=-40\\x\in \begin{bmatrix}\frac{10\left(5a_{0}+234\right)}{a_{0}+40},\frac{50a_{0}+311}{a_{0}}\end{bmatrix}\text{, }&a_{0}<-40\\x=\frac{50a_{0}+311}{a_{0}}\text{, }&a_{0}\geq \frac{12440}{29}\text{ or }a_{0}<0\\x\leq \frac{50a_{0}+311}{a_{0}}\text{, }&a_{0}>-40\text{ and }a_{0}<0\\x\in \begin{bmatrix}\frac{50a_{0}+311}{a_{0}},\frac{10\left(5a_{0}+234\right)}{a_{0}+40}\end{bmatrix}\text{, }&a_{0}\geq \frac{12440}{29}\\x=\frac{10\left(5a_{0}+234\right)}{a_{0}+40}\text{, }&a_{0}\leq -40\end{matrix}\right.$$

Solve for a_0

$$\left\{\begin{matrix}a_{0}\in \begin{bmatrix}\frac{20\left(2x-117\right)}{50-x},-\frac{311}{50-x}\end{bmatrix}\text{, }&x<50\\a_{0}\in \begin{bmatrix}-\frac{311}{50-x},\frac{20\left(2x-117\right)}{50-x}\end{bmatrix}\text{, }&x>50\text{ and }x\leq \frac{2029}{40}\end{matrix}\right.$$