Ninth Grade ยท Math ยท week 16

Systems of inequalities: where two regions overlap

Graph systems of linear inequalities and identify the feasible region; apply to simple constraint problems.

A system of inequalities asks for points that satisfy EVERY inequality at once. Graph each one's shaded half-plane the way you did last week, and the system's solution is wherever all the shadings overlap - often a wedge or polygon-shaped patch called the feasible region, because every point in it is a "feasible" (allowed) choice.

This is how real limits get modeled: a budget, a weight limit, a minimum requirement. Each constraint becomes one inequality; the feasible region is every combination that obeys all of them simultaneously. Later math (and real logistics/business decisions) often asks for the BEST point in that region - and that best point almost always sits at a corner where two boundary lines meet, not floating in the middle.

๐Ÿ“‹ For the grown-up teacher
Teach it (10โ€“15 min):
  • Shade one inequality in one color and the other in a second color on the same graph - the feasible region is where the colors overlap.
  • Ground it in a real constraint from home: a grocery budget split two ways, screen-time limits, a recipe that needs "at least" one ingredient.
  • Preview (lightly, no need to solve fully) that the best answer in these problems tends to live at a CORNER of the region.
  • Have them test one point that fails only ONE of the two inequalities - that near-miss is what makes the "at once" idea click.
Talk about it:
  • Why is the answer to a system of inequalities usually a whole region instead of one point?
  • What does it mean, physically, for a point to satisfy one constraint but not the other?
  • Why would the best (biggest profit, least cost) point tend to sit at a corner rather than the middle of the region?

Mastery looks like: They can test whether a point satisfies a system of two inequalities and set up a simple two-constraint system from a word description.

Common stumbles: Checking only one inequality and stopping; assuming "close" to satisfying a constraint (like 24 โ‰ค 25 needed but getting 24 when 25 was required) still counts.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open โ†’
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