Welcome to our comprehensive guide on understanding compound inequalities and how they are represented by graphs. In this article, we will delve into the concept of compound inequalities, their representation on graphs, and important examples for better comprehension.
Understanding Compound Inequalities
Compound inequalities are mathematical expressions that involve two or more inequalities connected together by the words “and” or “or”. These inequalities can be solved and represented graphically to visualize the range of values that satisfy the given conditions. In this article, we will focus on understanding compound inequalities involving “and”.
Representation on Graphs
When representing compound inequalities on graphs, it is important to understand how to interpret the graph and determine the range of values that satisfy the given conditions. The intersection of the individual inequalities represents the solution set for the compound inequality. Let’s take a look at an example to understand this concept in detail.
Example
Consider the compound inequality: -3 < x < 5. This compound inequality can be represented on a number line graph as follows:
| Number Line Graph |
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In this representation, the values of x that satisfy the compound inequality lie between -3 and 5, as indicated by the open circles at -3 and 5, and the solid line connecting them. This shows the range of values that x can take to satisfy the given compound inequality.
Identifying the Compound Inequality from a Graph
Now that we understand how compound inequalities are represented on graphs, let’s look at how to identify the compound inequality from a given graph. This process involves interpreting the graph and determining the appropriate inequalities that represent the shaded region on the graph. We will illustrate this with an example below.
Example 2
Consider the following number line graph representing a compound inequality:
| Number Line Graph |
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From the graph, we can see that the shaded region lies between -2 and 8, and includes -2 but not 8. This can be represented by the compound inequality: -2 ≤ x < 8.
Conclusion
Compound inequalities are essential concepts in mathematics that allow us to represent multiple conditions using inequalities. Understanding their representation on graphs is crucial for visualizing the range of values that satisfy the given compound inequality. With the examples and explanations provided in this article, we hope you now have a clear understanding of which compound inequality is represented by a given graph.
