Find the equivalent linear program

This is a classic question, its for a homework assignment, I have the solution but I've been breaking my head to understand how to approach the problem.

Minimize $$||A\vec x - \vec b||_1$$ subject to $$||x||_\infty \leq1$$

and the solution given is

Minimize $$\textbf{1}^T$$ subject to $$-y \leq A\vec x - \vec b \leq y$$ and $$\textbf{-1} \leq x \leq \textbf{1}$$

If anyone could shed some light on how to approach this problem...

The constraint $$\|x\|_\infty \le 1$$ means max of the coordinates of $$x$$ cannot be more than 1 in absolute value. How do you express this in terms of a regular inequality?
Also, think about what $$\|Ax-b\|_1$$ looks like if we have $$y = Ax-b$$?