Example 3:16. IncomeIn 2006, The Median Yearly Family Income Was About { $48,200$}$ Per Year. Suppose The Average Annual Rate Of Change Since Then Is { $1,240$}$ Per Year.a. Write And Graph An Inequality For The Annual Family
Understanding the Problem
In this example, we are given the median yearly family income in 2006, which was approximately $48,200 per year. We are also told that the average annual rate of change since then is $1,240 per year. Our goal is to write and graph an inequality that represents the annual family income.
Step 1: Define the Variables
Let's define the variables:
- x: the number of years since 2006
- y: the annual family income
Step 2: Write the Equation
Since the average annual rate of change is $1,240 per year, we can write the equation of the line that represents the annual family income as:
y = 48200 + 1240x
Step 3: Write the Inequality
We want to write an inequality that represents the annual family income. Since the average annual rate of change is $1,240 per year, we can write the inequality as:
y ≥ 48200 + 1240x
This inequality represents the annual family income that is greater than or equal to the median yearly family income in 2006, plus the average annual rate of change since then.
Step 4: Graph the Inequality
To graph the inequality, we can use a graphing calculator or a computer program. We can also use a table of values to help us graph the inequality.
| x | y |
|---|---|
| 0 | 48200 |
| 1 | 49440 |
| 2 | 50680 |
| 3 | 51920 |
| 4 | 53160 |
We can plot these points on a coordinate plane and draw a line through them. Since the inequality is greater than or equal to, we can shade the region above the line.
Step 5: Interpret the Graph
The graph represents the annual family income that is greater than or equal to the median yearly family income in 2006, plus the average annual rate of change since then. This means that any point above the line represents an annual family income that is higher than the median income in 2006.
Conclusion
In this example, we wrote and graphed an inequality that represents the annual family income. We defined the variables, wrote the equation, wrote the inequality, graphed the inequality, and interpreted the graph. This example demonstrates how to use mathematical concepts to model real-world problems.
Real-World Applications
This example has real-world applications in economics and finance. For example, policymakers can use this type of analysis to understand the trends in family income and make informed decisions about economic policies.
Mathematical Concepts
This example demonstrates the following mathematical concepts:
- Linear equations: We wrote a linear equation to represent the annual family income.
- Inequalities: We wrote an inequality to represent the annual family income that is greater than or equal to the median yearly family income in 2006, plus the average annual rate of change since then.
- Graphing: We graphed the inequality to visualize the relationship between the variables.
- Interpretation: We interpreted the graph to understand the meaning of the inequality.
Tips Variations
- Use different rates of change: We used a rate of change of $1,240 per year. You can try using different rates of change to see how it affects the graph.
- Use different initial values: We used an initial value of $48,200. You can try using different initial values to see how it affects the graph.
- Add more variables: You can add more variables to the equation to make it more complex.
- Use different types of inequalities: You can use different types of inequalities, such as strict inequalities or compound inequalities.
Example 3:16. Income Inequality Q&A =====================================
Q: What is the median yearly family income in 2006?
A: The median yearly family income in 2006 was approximately $48,200 per year.
Q: What is the average annual rate of change since 2006?
A: The average annual rate of change since 2006 is $1,240 per year.
Q: How do I write the equation of the line that represents the annual family income?
A: To write the equation of the line that represents the annual family income, you can use the formula:
y = 48200 + 1240x
Q: How do I write the inequality that represents the annual family income?
A: To write the inequality that represents the annual family income, you can use the formula:
y ≥ 48200 + 1240x
This inequality represents the annual family income that is greater than or equal to the median yearly family income in 2006, plus the average annual rate of change since then.
Q: How do I graph the inequality?
A: To graph the inequality, you can use a graphing calculator or a computer program. You can also use a table of values to help you graph the inequality.
Q: What does the graph represent?
A: The graph represents the annual family income that is greater than or equal to the median yearly family income in 2006, plus the average annual rate of change since then.
Q: What are some real-world applications of this example?
A: This example has real-world applications in economics and finance. For example, policymakers can use this type of analysis to understand the trends in family income and make informed decisions about economic policies.
Q: What are some mathematical concepts demonstrated in this example?
A: This example demonstrates the following mathematical concepts:
- Linear equations: We wrote a linear equation to represent the annual family income.
- Inequalities: We wrote an inequality to represent the annual family income that is greater than or equal to the median yearly family income in 2006, plus the average annual rate of change since then.
- Graphing: We graphed the inequality to visualize the relationship between the variables.
- Interpretation: We interpreted the graph to understand the meaning of the inequality.
Q: Can I use different rates of change or initial values?
A: Yes, you can use different rates of change or initial values to see how it affects the graph.
Q: Can I add more variables to the equation?
A: Yes, you can add more variables to the equation to make it more complex.
Q: Can I use different types of inequalities?
A: Yes, you can use different types of inequalities, such as strict inequalities or compound inequalities.
Conclusion
In this Q&A article, we answered some common questions about the example of the income inequality. We covered topics such as writing the equation and inequality, graphing the inequality, and interpreting the graph. We also discussed real-world applications and mathematical concepts demonstrated in the example