Graphing Inequalities — SAT Math Explained
Representing the solutions of a linear inequality on a coordinate plane as a shaded half-plane, with a boundary line separating solutions from non-solutions.
The Core Idea
A linear inequality divides the entire coordinate plane into two halves. One half satisfies the inequality, the other doesn't. The boundary line is the equation version of the inequality, and shading indicates the solution region.
Step-by-Step: How to Approach Graphing Inequalities
Rewrite the inequality as an equation and graph the boundary line
Determine line type: solid line for ≤ or ≥ (boundary included), dashed line for < or > (boundary excluded)
Pick a test point NOT on the line — (0, 0) is easiest unless the line passes through the origin
Substitute the test point into the original inequality
If the test point satisfies the inequality, shade the region containing it
If the test point does NOT satisfy it, shade the OTHER region
Solid Vs Dashed Line
Used for ≤ and ≥ — points ON the line are solutions
Used for < and > — points ON the line are NOT solutions
What Shading Means
Every point in the shaded region makes the inequality true when its (x, y) values are substituted in
Common Errors to Avoid
Drawing a solid line when the sign is strict (< or >)
Testing a point that's on the boundary line (it won't work as a test)
Shading the wrong half-plane after the test
How This Connects to Other Topics
The foundation for systems of inequalities and linear programming optimization problems
Practice: Graphing Inequalities
5 SAT-style questions. Select your answer and get an instant explanation.
When graphing y > 3x + 1 on the coordinate plane, what type of boundary line is drawn and which region is shaded?
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