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Graphing a Linear Inequality in Two Variables
To graph a linear inequality in two variables, follow a systematic three-step procedure:
Step 1 — Identify and graph the boundary line. Replace the inequality symbol with an equals sign to obtain the equation of the boundary line, then graph that line.
- If the inequality uses or , draw the boundary line as a solid line (the line is part of the solution set).
- If the inequality uses or , draw the boundary line as a dashed line (the line is not part of the solution set).
Step 2 — Test a point that is not on the boundary line. Choose any point that does not lie on the boundary line and substitute its coordinates into the original inequality. The origin is usually the most convenient choice because the arithmetic is simple. However, if the boundary line passes through the origin, cannot be used and a different point must be selected.
Step 3 — Shade the correct side of the boundary line.
- If the test point is a solution (the inequality is true), shade the half-plane that contains the test point.
- If the test point is not a solution (the inequality is false), shade the half-plane on the opposite side.
The shaded region, together with the boundary line (if solid), represents the complete solution set — every ordered pair in that region satisfies the inequality.
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