Using a Test Point to Determine Shading
Using a test point to determine shading is a Grade 7 math skill from Yoshiwara Intermediate Algebra applied when graphing linear inequalities. After drawing the boundary line, students substitute a test point (usually the origin) into the inequality to determine which half-plane to shade.
Key Concepts
Property To find which half plane contains the solutions (and should be shaded), use the Test Point Method: 1. Graph the boundary line (solid or dashed). 2. Choose a simple test point that is strictly NOT on the boundary line. The origin $(0, 0)$ is always the best choice, unless the line passes directly through it. 3. Substitute the coordinates of the test point into the original inequality. 4. If the result is a TRUE statement, shade the entire half plane that contains the test point. If it is FALSE, shade the opposite half plane.
Examples Using $(0,0)$ as a test point: Graph $x 3y < 6$. Draw the dashed line $x 3y = 6$. Test the origin $(0,0)$: $0 3(0) < 6$ simplifies to $0 < 6$. This is a TRUE statement. Therefore, shade the side of the line that includes the point $(0,0)$. When $(0,0)$ is on the line: Graph $y \leq 2x$. Draw the solid line $y = 2x$. Since this line passes exactly through the origin, we must choose a different point, like $(3, 1)$. Test it: $1 \leq 2(3)$ simplifies to $1 \leq 6$. This is TRUE. Shade the side containing the point $(3, 1)$. A False Result: Graph $y 4$. Draw a dashed horizontal line at $y = 4$. Test $(0,0)$: $0 4$ is FALSE. Shade the side that does NOT contain $(0,0)$, which is the region above the line.
Explanation Because the boundary line cuts the graph perfectly in half, all the correct answers live together on one side, and all the wrong answers live together on the other. This means you don't have to test a hundred different points! You only need to test one single point to scout out the territory. If your scout point tells the truth, its entire side is the winner. If it lies, the other side wins.
Common Questions
How do you use a test point to shade a linear inequality graph?
After drawing the boundary line, pick a point not on the line (usually (0,0)). Substitute it into the inequality. If it makes the inequality true, shade the region containing that point. If false, shade the other side.
Why is (0,0) the most common test point?
(0,0) makes arithmetic very easy. Just be sure the boundary line does not pass through the origin, or choose a different test point.
What does the shaded region represent?
The shaded region represents all (x, y) pairs that satisfy the inequality — all solutions to the system.
Does a solid or dashed boundary line mean the boundary is included?
A solid line means ≤ or ≥ (boundary included). A dashed line means < or > (boundary not included).