Ninth Grade · Math · week 15

Linear inequalities: shading a half-plane

Graph linear inequalities in two variables using a boundary line and a test point.

An equation like y = 2x + 3 draws one line. An inequality like y < 2x + 3 describes an entire region - every point below that line. The boundary line itself is drawn dashed for strict inequalities (< or >), since points ON the line do not satisfy them, and solid for ≤ or ≥, since the line's own points count as solutions.

To know WHICH side to shade, pick any test point not on the line - the origin (0, 0) is usually easiest - and substitute it into the inequality. If it makes the inequality true, shade the side containing that point; if false, shade the other side. This works because crossing the boundary line is the only place the inequality's truth value can flip.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Draw the boundary line together first, ignoring the inequality sign completely - treat it as an ordinary y = mx + b line.
  • Make solid-vs-dashed a physical rule they say out loud every time: "does the line count as an answer, or not?"
  • Always test (0, 0) unless the line passes through the origin - then pick an easy point like (1, 0) or (0, 1) instead.
  • Have them plug the test point into the ORIGINAL inequality, not a rearranged version, to avoid sign confusion.
Talk about it:
  • Why does a point on one side of the line always give the same true/false result as any other point on that side?
  • When would you NOT be able to use (0, 0) as your test point?
  • What is the difference between the boundary line and the actual solution set?

Mastery looks like: They can graph a linear inequality with correctly solid or dashed boundary and correctly shaded side, explaining the test-point logic.

Common stumbles: Reversing the inequality direction has nothing to do with shading (that only matters when dividing by a negative); shading the wrong side after a correct true/false test.

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