Questions on linear programming, the optimization of a linear function subject to linear constraints.

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Convert a piecewise linear function into a linear optimisation problem.

Consider $$f(x) = \left\{\begin{matrix} 1-x, & 0 \le x < 1\\ x-1, & 1 \le x < 2\\ \frac{x}{2}, & 2 \le x \le 3 \end{matrix}\right.$$ where $x \ge 0$. Convert $$\min z = ...
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When might some a variable leave the basis?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...