The wavy curve method is a visual technique used in graphing inequalities. It helps you determine the solution set for an inequality by identifying the regions on a coordinate plane that satisfy the inequality.
How it Works:
- Graph the boundary line: First, you graph the equation that forms the boundary of the inequality. If the inequality includes "less than or equal to" or "greater than or equal to," the line is solid. If the inequality includes "less than" or "greater than," the line is dashed.
- Choose a test point: Select any point on the coordinate plane that is not on the boundary line.
- Substitute the test point: Substitute the coordinates of the test point into the original inequality.
- Determine the solution region:
- If the test point satisfies the inequality, shade the region of the plane containing the test point.
- If the test point does not satisfy the inequality, shade the region of the plane not containing the test point.
Example:
Graph the inequality: y < 2x + 1
- Graph the boundary line: The equation of the boundary line is y = 2x + 1. Graph this line as a dashed line because the inequality is "less than."
- Choose a test point: Let's choose the origin (0, 0).
- Substitute the test point: Substitute (0, 0) into the inequality: 0 < 2(0) + 1. This simplifies to 0 < 1, which is true.
- Determine the solution region: Since the test point satisfies the inequality, shade the region of the plane containing the origin.
Practical Insights:
- The wavy curve method is a simple and effective way to visualize the solution set of an inequality.
- It is particularly useful for understanding inequalities with two variables.
- This method can be applied to solve real-world problems involving inequalities, such as finding the optimal production level for a company.
