Question

hecking account earnings at Long's Bank are expressed by the equation, I=−0.06x+8.3
.

The earnings at Fellow's Bank are modeled by I=−0.02x+6.6
.

In both cases, x is the number of checks written.

For what range of checks will a checking account at Long's Bank generate more earnings income than one at Fellow's Bank? Show your work or explain how you got your answer.

(4 points)

Answer 1

To determine the range of checks for which a checking account at Long's Bank generates more earnings than one at Fellow's Bank, we need to set their earnings equations equal to each other and solve for \(x\).

The earnings equations are:
- Long's Bank: \( I_L = -0.06x + 8.3 \)
- Fellow's Bank: \( I_F = -0.02x + 6.6 \)

We want to find the range of \(x\) where:
\[ I_L > I_F \]

This leads us to the inequality:
\[ -0.06x + 8.3 > -0.02x + 6.6 \]

To solve this inequality, follow these steps:

1. **Rearrange the terms:**
Subtract \(-0.02x\) and \(6.6\) from both sides:
\[
-0.06x + 0.02x > 6.6 - 8.3
\]
This simplifies to:
\[
-0.04x > -1.7
\]

2. **Divide by -0.04:**
Remember that when you divide an inequality by a negative number, you need to reverse the inequality sign:
\[
x < \frac{-1.7}{-0.04}
\]
Calculate the right side:
\[
x < 42.5
\]

So far, we have found that \( x < 42.5 \).

3. **Checking for lower bounds:**
The number of checks, \(x\), cannot be negative since it represents the number of checks written. Therefore, we also have:
\[
x \geq 0
\]

Combining the results, we find the range of \(x\):
\[
0 \leq x < 42.5
\]

**Final Result:**
The checking account at Long's Bank will generate more earnings income than one at Fellow's Bank when the number of checks written is in the range of:
\[
\boxed{[0, 42.5)}
\]