key term - Dashed Line

Definition

A dashed line is a type of line that is composed of a series of short line segments separated by small gaps, creating a discontinuous visual effect. This line style is commonly used in various graphical representations, including the graphing of linear inequalities and systems of linear inequalities.

5 Must Know Facts For Your Next Test

1. Dashed lines are used to represent the boundary or solution set of a linear inequality on a graph.
2. The dashed line indicates that the boundary is not part of the solution set, as it represents a strict inequality (< or >).
3. In a system of linear inequalities, each inequality is represented by a dashed line, and the feasible region is the area where all the inequalities are satisfied.
4. The shaded region on a graph of a linear inequality or a system of linear inequalities represents the solution set, which is the area where the inequality or system of inequalities is true.
5. The intersection of the shaded regions for a system of linear inequalities represents the feasible region, which is the set of all possible solutions.

Review Questions

• Explain the purpose of using a dashed line to represent the boundary of a linear inequality on a graph.
  • The dashed line is used to represent the boundary of a linear inequality on a graph to indicate that the boundary itself is not part of the solution set. This is because a dashed line represents a strict inequality, such as $x < 3$ or $y > 2$, where the boundary values ($x = 3$ or $y = 2$) are not included in the solution set. The shaded region on the graph represents the actual solution set for the linear inequality.
• Describe how dashed lines are used to graph a system of linear inequalities and how the feasible region is determined.
  • When graphing a system of linear inequalities, each individual inequality is represented by a dashed line on the graph. The feasible region is the area where all the inequalities are satisfied, which is the intersection of the shaded regions for each inequality. The dashed lines indicate the boundaries of the inequalities, and the shaded regions represent the solution sets for each individual inequality. The feasible region is the area where all the shaded regions overlap, representing the set of all possible solutions to the system of linear inequalities.
• Analyze the role of dashed lines in the context of linear inequalities and systems of linear inequalities, and explain how they contribute to the understanding and interpretation of the solution sets.
  • Dashed lines play a crucial role in the graphical representation of linear inequalities and systems of linear inequalities. They serve to distinguish the boundary of the inequality from the actual solution set, which is the shaded region on the graph. The dashed line indicates that the boundary values are not included in the solution set, as they represent a strict inequality. This visual distinction helps students understand the difference between the boundary and the solution set, and how to interpret the graph to determine the valid solutions. Furthermore, in the context of systems of linear inequalities, the dashed lines for each individual inequality contribute to the identification of the feasible region, which is the area where all the inequalities are satisfied. The intersection of the shaded regions, bounded by the dashed lines, represents the set of all possible solutions to the system of linear inequalities.

Related terms

Solid Line:

A solid line is a continuous, uninterrupted line that is commonly used to represent the boundary or solution set of an equation or inequality.

Shaded Region:

The shaded region is the area on a graph that represents the solution set of a linear inequality or a system of linear inequalities.

Feasible Region:

The feasible region is the area on a graph that satisfies all the constraints of a system of linear inequalities, representing the set of all possible solutions.