Using an exponential function, the inequality is given as follows:
![9400(0.857)^t < 6000](https://tex.z-dn.net/?f=9400%280.857%29%5Et%20%3C%206000)
The solution is t > 2.9, hence the tax status will change within the next 3 years.
<h3>What is an exponential function?</h3>
A decaying exponential function is modeled by:
![A(t) = A(0)(1 - r)^t](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A%280%29%281%20-%20r%29%5Et)
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal.
For this problem, the parameters are given as follows:
A(0) = 9400, r = 0.143.
The population after t years is modeled by:
![A(t) = A(0)(1 - r)^t](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A%280%29%281%20-%20r%29%5Et)
![A(t) = 9400(1 - 0.143)^t](https://tex.z-dn.net/?f=A%28t%29%20%3D%209400%281%20-%200.143%29%5Et)
![A(t) = 9400(0.857)^t](https://tex.z-dn.net/?f=A%28t%29%20%3D%209400%280.857%29%5Et)
The tax status will change when:
![A(t) < 6000](https://tex.z-dn.net/?f=A%28t%29%20%3C%206000)
Hence the inequality is:
![9400(0.857)^t < 6000](https://tex.z-dn.net/?f=9400%280.857%29%5Et%20%3C%206000)
Then:
![(0.857)^t < \frac{6000}{9400}](https://tex.z-dn.net/?f=%280.857%29%5Et%20%3C%20%5Cfrac%7B6000%7D%7B9400%7D)
![\log{(0.857)^t} < \log{\left(\frac{6000}{9400}\right)}](https://tex.z-dn.net/?f=%5Clog%7B%280.857%29%5Et%7D%20%3C%20%5Clog%7B%5Cleft%28%5Cfrac%7B6000%7D%7B9400%7D%5Cright%29%7D)
![t\log{0.857} < \log{\left(\frac{6000}{9400}\right)}](https://tex.z-dn.net/?f=t%5Clog%7B0.857%7D%20%3C%20%5Clog%7B%5Cleft%28%5Cfrac%7B6000%7D%7B9400%7D%5Cright%29%7D)
Since both logs are negative:
![t > \frac{\log{\left(\frac{6000}{9400}\right)}}{\log{0.857}}](https://tex.z-dn.net/?f=t%20%3E%20%5Cfrac%7B%5Clog%7B%5Cleft%28%5Cfrac%7B6000%7D%7B9400%7D%5Cright%29%7D%7D%7B%5Clog%7B0.857%7D%7D)
t > 2.9.
The solution is t > 2.9, hence the tax status will change within the next 3 years.
More can be learned about exponential functions at brainly.com/question/25537936
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