An inequality with a symbol is graphed with a solid line and includes the points on the line as part of the solution.
An inequality with a symbol is graphed with a dashed line and excludes the points on the line as part of the solutions.
PART I: When graphing an inequality with a symbol using a solid line, it indicates that the points on the line are included as part of the solution set. In mathematical terms, this solid line represents a "less than or equal to" (≤) or "greater than or equal to" (≥) inequality. The solid line emphasizes that the values lying directly on the line satisfy the given inequality. The equality component is crucial here, indicating that the points on the line are valid solutions and contribute to the shaded region representing the solution set.
For example, if you have the inequality y≤2x+3, graphing it with a solid line would signify that all points on the line y=2x+3 and below it are solutions to the inequality.
PART II: Conversely, when an inequality is graphed with a dashed line, it signifies that the points on the line are not included as part of the solution set. This is commonly used for "less than" (<) or "greater than" (>) inequalities. The dashed line indicates a strict inequality where the values on the line itself do not satisfy the given inequality. The shaded region, therefore, represents all points either greater than or less than the line, depending on the specific inequality.
For instance, if you have the inequality y>2x−1, a dashed line would be drawn to show that the points on the line y=2x−1 are not solutions, and the shaded region would extend only to points greater than the line. This visual representation is essential in understanding the nature of the solution set and the inclusion or exclusion of points on the boundary.