Mike had $13,500 in his checkings account at the beginning of year 2021. Each year, the net value of his account increases by $8,400. If the value of the checkings account $t$ years after 2021 was between $33,500 and $43,000, which of the following must be true?

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The amount $A$ in the checkings account $t$ years after 2021 can be modeled by a simple linear equation:

$$A=8{,}400t+13{,}500$$

We are trying to find the value of $t$ when the amount is between $33{,}500$ and $43{,}000$.

$$33{,}500 \lt A \lt 43{,}000$$ $$33{,}500 \lt 8{,}400t+13{,}500 \lt 43{,}000$$ $$20{,}000 \lt 8{,}400t \lt 26{,}500$$ $$2.38 \lt t \lt 3.51$$

The range $\boxed{2 \lt t \lt 4}$ includes all values from our inequality above.

We can calculate the amount in the checkings account in the first few years.

$$\begin{array}{|c|c|} \hline \text{$t$} & \text{amount} \\ \hline 0 & \$13{,}500 \\ \hline 1 & \$21{,}900 \\ \hline 2 & \$30{,}300 \\ \hline 3 & \$38{,}700 \\ \hline 4 & \$47{,}100 \\ \hline \end{array}$$

The given amounts in our checkings account are $33,500 and $43,000, which are between $t=2$ and $t=4$.