lesson
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Imagine you have a 20giftcardtospendonsnacks(2 each) and drinks (4each).Youdon′thavetospendeverypenny—youjustcannotspendmorethan20.
An equation like 2x+4y=20 only shows combinations that cost exactly 20. To show every affordable combination, we graph an inequality and shade an entire region.
The Boundary Line
Every linear inequality in two variables starts with a boundary line, which cuts the coordinate plane into two infinite halves called half-planes.
If the inequality uses ≤ or ≥, the boundary line is solid because points directly on the line are included as solutions. If it uses < or >, the line is dashed because points on the line are excluded.
📊A visual comparison card showing two side-by-side coordinate planes with lines. Card 1: Line labeled 'y ≤ 2x + 1' drawn as a solid blue line with a badge 'Solid: Includes the line (≤, ≥)'. Card 2: Line labeled 'y < 2x + 1' drawn as a dashed orange line with a badge 'Dashed: Excludes the line (<, >)'. Clean modern look, light background (#f8fafc), text #1e2945, border #e2e8f0, responsive width.
How do you know which of the two half-planes contains the solutions? Let's look at the most reliable technique.
The Test Point Strategy
To find which side to shade, choose any test point not on the boundary line and substitute its coordinates into the inequality.
The origin (0,0) is almost always the easiest test point because multiplying numbers by zero simplifies calculations instantly.
If the test point produces a true statement, shade the half-plane containing that point; if false, shade the opposite half-plane.
Step-by-Step Example
Let's graph the inequality y<2x−1.