To graph inequalities, follow these clear steps:
-
Graph the boundary line :
Replace the inequality symbol (<, >, ≤, ≥) with an equals sign (=) and graph the resulting equation as if it were a line.- Use a dashed line if the inequality is strict (< or >), meaning the points on the line are not included in the solution.
- Use a solid line if the inequality includes equality (≤ or ≥), meaning the points on the line are included in the solution
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Choose a test point :
Pick a point not on the boundary line (commonly (0,0) if it is not on the line) and substitute its coordinates into the original inequality.- If the inequality holds true for this point, shade the region of the graph that contains this point.
- If it does not hold true, shade the opposite side
- Shade the solution region :
The shaded area represents all the points that satisfy the inequality. This region lies on one side of the boundary line.
Example
Graph the inequality y>2x+3y>2x+3y>2x+3:
- Step 1: Graph the line y=2x+3y=2x+3y=2x+3 with a dashed line because the inequality is strict (>).
- Step 2: Test the point (0,0): 0>2(0)+3⇒0>30>2(0)+3\Rightarrow 0>30>2(0)+3⇒0>3 is false.
- Step 3: Since (0,0) is not a solution, shade the region on the opposite side of the line from (0,0)
This method works for all linear inequalities in two variables and can be extended to absolute value inequalities and systems of inequalities by graphing multiple boundary lines and shading the intersection of solution regions
. In summary, graphing inequalities involves drawing the boundary line (dashed or solid depending on inclusion), testing a point to determine which side to shade, and shading the region that satisfies the inequality.
