After a ten year study we find that two towns, A and B, have populations modeled by f(x)=-350x+20,000 and f(x)=250x+10,000 respectively, where x is the current year. For what years will the population of town B be greater than the population of town A? (Numbers may be rounded to 2 decimal places.)

To determine the years during which the population of town B will be greater than the population of town A, we need to set up an inequality based on the given population models.

The population models for towns A and B are given by:
\[ f_A(x) = -350x + 20,000 \]
\[ f_B(x) = 250x + 10,000 \]

We need to find the values of \( x \) for which \( f_B(x) > f_A(x) \).

Set up the inequality:
\[ 250x + 10,000 > -350x + 20,000 \]

Combine like terms:
\[ 250x + 350x > 20,000 - 10,000 \]
\[ 600x > 10,000 \]

Solve for \( x \):
\[ x > \frac{10,000}{600} \]
\[ x > \frac{1000}{60} \]
\[ x > \frac{100}{6} \]
\[ x > 16.67 \]

Thus, the population of town B will be greater than the population of town A for any year \( x \) greater than 16.67. Assuming \( x = 0 \) corresponds to the current year, the population of town B will surpass that of town A starting in year \( x = 17 \).

### Summary
The population of town B will be greater than the population of town A starting in the 17th year from now. If we consider the current year as \( x = 0 \), this will occur 17 years from now.

### Questions for Further Exploration
1. What will the population difference be exactly at year 17?
2. How does the rate of change in population affect long-term trends for both towns?
3. In what year will the population of town A be half that of town B?
4. How would the results change if the population trends were reversed?
5. What other factors could influence the population growth of these towns that aren't accounted for in these linear models?

### Tip
When working with linear equations and inequalities, always combine like terms and isolate the variable of interest to solve for it accurately.

Explore a population comparison between two towns, A and B, using linear growth models. Learn how to set up and solve inequalities to determine when the population of town B will surpass that of town A. Suitable for students in Grades 9-11.

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