Question

If the graph of a system of inequalities cannot be contained in any circle, then the graph is: (a) bounded (b) unbounded (c) decomposed (d) composed

Short answer

The graph is unbounded (b).

Step-by-step solution

- Understanding the Graph

Begin by visualizing what it means for a graph to be contained within a circle. If the graph can fit inside some circle, it means that the graph is limited in extent in all directions.

- Analyze the Term 'Bounded'

A graph is described as 'bounded' if its extent is limited; in other words, it can be enclosed within some circle.

- Analyze the Meaning of 'Unbounded'

A graph is described as 'unbounded' if it extends infinitely in at least one direction and cannot be enclosed within any circle.

- Consider Why the Graph Cannot Be Contained in Any Circle

The statement specifies the graph of the system of inequalities cannot be contained within any circle, indicating the graph extends infinitely.

- Determine the Correct Answer

Based on the analysis, since the graph cannot be contained in any circle, it must be unbounded.

Key concepts

bounded graphs

When discussing systems of inequalities, we often visualize their solutions as graphs. Sometimes, these graphs can be confined within a boundary. This concept is known as a bounded graph. A bounded graph is one where all points fit neatly inside a circle or any other closed shape. In mathematical terms, we say that the graph is limited in every direction.

For example, consider the inequalities:
\[ x + y \leq 4 \] \text{ and } \[ x - y \geq -2 \].
Plotting these on a graph, you would notice that the line segments and points where these inequalities overlap do not stretch out to infinity. Instead, they form a closed shape which forms a bounded region.

Practical applications of bounded graphs are numerous. They appear in resource allocation problems, where resources (like time or money) are limited and thus represented within a finite region.

unbounded graphs

In contrast to bounded graphs, unbounded graphs extend indefinitely in at least one direction. They cannot be contained within any circle or finite boundary.

Let's take an example of an unbounded graph with the inequality:
\[ x \geq 2 \].
When this inequality is plotted, you'll notice the solution area extends infinitely to the right and never caps off. This unbound nature means no matter how large a circle you draw, the graph will always have some part outside of it.

Situations where unbounded graphs come into play include many real-world scenarios such as economic models where certain variables (like potential profit) can grow indefinitely without restriction.

graph analysis

Graph analysis is a crucial skill when dealing with systems of inequalities. It includes understanding the shape and limits of the graph to determine whether it is bounded or unbounded.
To analyze a graph effectively, follow these steps:

• Identify the inequalities and plot each one on the coordinate system.
• Determine the region where all inequalities overlap.
• Assess whether this overlapping region can be contained within a circle or not.

If the region can be entirely enclosed within some circle, then the graph is bounded. If not, the graph is unbounded.

Analyzing graphs helps in fields like physics, economics, and engineering, where understanding the constraints and limits of a system is vital for making accurate predictions and decisions.