Eleventh Grade · Math · week 7

Systems of inequalities and linear programming

Graph systems of linear inequalities and use the vertices of the feasible region to solve a linear programming (optimization) problem.

A linear inequality like y <= 2x+3 does not describe a line -- it describes an entire half of the plane, with the line itself as the boundary. Graph the boundary line (solid if the inequality includes equals, dashed if it does not), then test one point NOT on the line to decide which half to shade. A system of inequalities is the OVERLAP of every shaded region -- the feasible region -- since a point must satisfy every constraint at once.

Linear programming asks a very practical question: given constraints (limited hours, limited materials, limited budget), what mix of choices maximizes profit or minimizes cost? The key fact that makes this solvable by hand: the maximum or minimum of a linear objective function always occurs at a VERTEX (corner point) of the feasible region -- never in the middle. So the whole method is: find the region, find its corners, plug each corner into the objective function, and pick the best.

📋 For the teacher
Teach it (10–15 min):
  • Graph one inequality at a time with an actual test point (usually (0,0), unless the line passes through the origin) -- "plug in and check" beats trying to memorize which way to shade.
  • Track the OVERLAP only -- a common error is shading each inequality's region separately and forgetting only the intersection counts as feasible.
  • Make the vertex-checking step mechanical: list every vertex first, THEN evaluate the objective function at each one in a little table, so nothing gets skipped.
  • Ground it in a real decision: a small business choosing how many of two products to make with limited hours and materials -- optimization is what businesses actually do with this math.
Talk about it:
  • Why is the feasible region the OVERLAP of the shaded regions, not the combination of all of them?
  • Why does the best answer always land on a corner instead of somewhere in the middle of the region?
  • How would you find the coordinates of a corner that is not on either axis?

Mastery looks like: They correctly identify the feasible region of a system of inequalities, list its vertices, and evaluate a linear objective function to find the maximum or minimum.

Common stumbles: Shading the wrong side of a boundary line; checking only some of the vertices instead of every one; forgetting that a corner off the axes needs its own system to be solved.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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