Shading inequalities on a graph depends on the inequality symbol used.
Understanding Inequality Symbols
The inequality symbol determines which side of the line to shade:
- Greater than (>) or greater than or equal to (≥): Shade above the line.
- Less than (<) or less than or equal to (≤): Shade below the line.
Shading Rules
Here's a simple way to remember the shading rules:
| Inequality Symbol | Shading Direction |
|---|---|
| > (Greater than) | Above the line |
| ≥ (Greater than or equal to) | Above the line |
| < (Less than) | Below the line |
| ≤ (Less than or equal to) | Below the line |
Practical Examples
Let's illustrate with some examples:
- Example 1: y > 2x + 1 - Since this is a "greater than" inequality, you would shade the area above the line y = 2x + 1.
- Example 2: y ≤ -x + 3 - Here, we have "less than or equal to," so you would shade the area below the line y = -x + 3.
- Example 3: x > 4 - Since this is greater than, you would shade to the right of the vertical line x = 4.
- Example 4: y < -2 - Since this is less than, you would shade below the horizontal line y = -2.
Key Considerations
- Solid vs. Dashed Lines: Inequalities with "greater than or equal to" (≥) or "less than or equal to" (≤) use a solid line. Inequalities with just "greater than" (>) or "less than" (<) use a dashed line. The dashed line indicates that the points on the line are not part of the solution set.
- Test Point: If you are unsure which side of the line to shade, you can select a test point not on the line. Substitute the x and y coordinates of your test point into the inequality. If the inequality is true, shade the side of the line where the test point is located. If the inequality is false, shade the other side. A simple test point to use is (0, 0) whenever the line does not pass through the origin.
- Understanding the Solution Set: The shaded region represents all the points that satisfy the inequality. These are the solutions.
By understanding these rules, you can confidently shade any inequality on a graph. Remember to always consider the inequality symbol and whether the line should be solid or dashed.
