Solve {l}{ax-1leq0}{x-4ageq0}

$$\left\{ \begin{array} { l } { a x - 1 \leq 0 } \\ { x - 4 a \geq 0 } \end{array} \right.$$

Solve for x

$$\left\{\begin{matrix}x\geq 0\text{, }&a=0\\x\geq 4a\text{, }&a\geq -\frac{1}{2}\text{ and }a<0\\x\geq \frac{1}{a}\text{, }&a<-\frac{1}{2}\\x\in \begin{bmatrix}4a,\frac{1}{a}\end{bmatrix}\text{, }&a>0\text{ and }a\leq \frac{1}{2}\\x=\frac{1}{a}\text{, }&a\leq -\frac{1}{2}\text{ or }\left(a\geq 0\text{ and }a\leq \frac{1}{2}\right)\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a\leq 0\text{, }&x=0\\a\in \begin{bmatrix}\frac{1}{x},\frac{x}{4}\end{bmatrix}\text{, }&x\geq -2\text{ and }x<0\\a\leq \frac{x}{4}\text{, }&x>0\text{ and }x\leq 2\\a\leq \frac{1}{x}\text{, }&x>2\\a=\frac{1}{x}\text{, }&\left(x\geq -2\text{ and }x\leq 0\right)\text{ or }x\geq 2\end{matrix}\right.$$