Systems of linear inequalities
A system of linear inequalities describes all the points that satisfy two or more inequalities at once. To solve it graphically, graph each boundary line (dashed for < or >, solid for <= or >=), shade the side that makes each inequality true, and the solution set is the region where all the shadings overlap. Any point in that region works, and you can check a point by substituting it into every inequality. Systems like this model real constraints: if tickets cost $2 for students (x) and $5 for adults (y), a club needs at least $20 in sales, and the room holds at most 10 people, then x + y <= 10, 2x + 5y >= 20, x >= 0, and y >= 0. The point (2, 4) works, since 2 + 4 = 6 <= 10 and 2(2) + 5(4) = 24 >= 20, but (1, 1) does not, since 2 + 5 = 7 is less than 20.
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