Using a test point
Use a test point to determine which side of a boundary line satisfies an inequality: substitute any point not on the line and shade the side where the inequality is true.
Key Concepts
To determine which half plane contains the solutions, use a test point not on the boundary line. If the test point satisfies the inequality, shade that half plane. The point $(0, 0)$ is a good test point when available.
For $3y + x \geq 9$, test $(0, 0)$: $3(0) + 0 \geq 9 \rightarrow 0 \geq 9$. This is true, so shade the half plane containing $(0, 0)$. For $y 4x + 2$, test $(0, 0)$: $0 4(0) + 2 \rightarrow 0 2$. This is false, so shade the half plane that does not contain $(0, 0)$.
Think of the test point as a secret agent. You send the easiest agent, $(0, 0)$, into one of the half planes. If it reports back 'True,' that whole region is the solution, and you shade it in! If it reports 'False,' you know the solutions are all hiding in the other half plane, so you shade that one instead.
Common Questions
How do you use a test point when graphing a linear inequality?
After graphing the boundary line, choose any point not on the line, typically (0,0) if it is not on the line. Substitute the point's coordinates into the inequality. If the resulting statement is true, shade the region containing that test point. If false, shade the opposite region.
What test point should you use when the line passes through the origin?
When the boundary line passes through the origin, (0,0) lies on the line and cannot be used as a test point. Instead choose any other convenient point such as (1,0), (0,1), or (1,1), substitute into the inequality, and shade accordingly.
Why does the test point method work for any linear inequality?
A linear inequality divides the plane into exactly two half-planes, one satisfying the inequality and one not. Because the boundary is a line (not a curve), any single point not on the line correctly identifies its entire half-plane. Testing any one interior point is sufficient.