Graphical inequalities are visual representations of mathematical inequalities on a coordinate plane, showing the set of all points that satisfy the inequality. They are displayed as a shaded region bounded by a line (or curve), where the line indicates the boundary of the solution set.
Understanding the Components
- Inequality: A mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
- Coordinate Plane: A two-dimensional plane formed by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), used to plot points.
- Boundary Line: The line representing the equation formed by replacing the inequality symbol with an equals sign (=). This line separates the coordinate plane into two regions.
- Shaded Region: The area of the coordinate plane that contains all the points whose coordinates satisfy the inequality. This represents the solution set of the inequality.
- Dashed vs. Solid Line: A dashed line indicates that the points on the line are not included in the solution set (used for < and >), while a solid line indicates that the points on the line are included in the solution set (used for ≤ and ≥).
How to Graph an Inequality
- Replace the inequality symbol with an equals sign and graph the resulting equation. This gives you the boundary line.
- Determine if the boundary line should be solid or dashed. Use a solid line for ≤ or ≥, and a dashed line for < or >.
- Choose a test point that is NOT on the line. A common choice is (0, 0) if the line doesn't pass through the origin.
- Substitute the coordinates of the test point into the original inequality.
- If the inequality is true, shade the region that contains the test point. If the inequality is false, shade the region that does not contain the test point.
Example
Consider the inequality y > 2x + 1.
- Graph the line:
y = 2x + 1. This is a line with a slope of 2 and a y-intercept of 1. - Dashed or Solid? Since the inequality is
>, the line should be dashed. - Choose a test point: Let's use (0, 0).
- Substitute:
0 > 2(0) + 1becomes0 > 1, which is false. - Shade: Because (0, 0) made the inequality false, we shade the region that does not contain (0, 0). This is the region above the dashed line.
More Complex Inequalities
Graphical inequalities can also involve more complex functions than just lines, such as parabolas, circles, and other curves. The same principles apply: graph the boundary curve, determine if it's solid or dashed, and use a test point to determine which region to shade.
