Problem 91 - College Algebra Essentials

Question

What does a dashed line mean in the graph of an inequality?

Step-by-Step Solution

Answer

A dashed line in the graph of an inequality indicates that the points on the line are not included in the solution set of the inequality. It usually signifies a 'greater than' or 'less than' inequality.

Step 1: Understanding Inequalities

Inequalities are mathematical expressions involving the symbols '<', '>', '≤' or '≥'. They show the relationship between two expressions that may not be equal. These symbols mean 'less than', 'greater than', 'less than or equal to' and 'greater than or equal to' respectively.

Step 2: Graphical Representation of Inequalities

In a graph, inequalities are represented by a straight line, which could be either solid or dashed. The type of the line shows the nature of the inequality.

Step 3: Dashed Line in Inequalities

A dashed or dotted line in the graph of an inequality usually signifies that the corresponding linear inequality is 'greater than' or 'less than'. It shows that the points on the line are not included in the solution of the inequality.

Step 4: Example

For instance, let's take the inequality \( y < 2x + 3 \). Here, the line \( y = 2x + 3 \) is represented on the graph as a dashed line, because the inequality symbol is '<', demonstrating that the y-values are less than those along \( y = 2x + 3 \), but do not include the actual values on the line.