• Adding or subtracting the same number on both sides keeps the inequality sign.
• Multiplying or dividing by a negative number reverses the sign.
• Compound inequality: $a < x \le b$ means both $x > a$ and $x \le b$.

Graphing $y \ge mx + c$ (or similar)

1. Draw the boundary line $y = mx + c$: solid for $\le$ or $\ge$, dashed for $<$ or $>$.
2. Test a point (e.g. the origin) to decide which side to shade (or leave unshaded, as the question asks).

Linear programming

1. Define the variables ($x$, $y$) and write the constraints as inequalities.
2. Write the objective function, e.g. $P = ax + by$ (profit to maximise or cost to minimise).
3. Graph the region that satisfies all constraints.
4. Evaluate the objective function at each vertex of the region, or slide a search line; the best vertex gives the optimum.
5. Use whole numbers when the variables are items (people, machines...).