Solve {l}{2|x-a|+a^2>y}{xgeq1}{ygeq4}

$$\left. \begin{array} { l } { 2 | x - a | + a ^ { 2 } > y } \\ { x \geq 1 } \\ { y \geq 4 } \end{array} \right.$$

Solve for x

$$\left\{\begin{matrix}x\in [1,\frac{a^{2}}{2}-\frac{y}{2}+a)\text{, }&y\geq 4\text{ and }y<a^{2}+2a-2\\x\geq 1\text{, }&(y\geq 4\text{ and }y<a^{2}-2a+2)\text{ or }(y\geq 4\text{ and }y<a^{2})\\x>-\frac{a^{2}}{2}+\frac{y}{2}+a\text{, }&y\geq a^{2}-2a+2\text{ and }y\geq 4\end{matrix}\right.$$

Solve for y

$$y\in [4,a^{2}+2|x-a|)\text{, }(a=-2\text{ and }x\geq 1)\text{ or }(x\neq 2\text{ and }a=2\text{ and }x\geq 1)\text{ or }(x>-\frac{a^{2}}{2}+a+2\text{ and }a>-2\text{ and }x\geq 1\text{ and }|a|<2)\text{ or }(x<\frac{a^{2}}{2}+a-2\text{ and }a>-2\text{ and }x\geq 1\text{ and }|a|<2)\text{ or }(x\geq 1\text{ and }|a|>2)$$