Question

To determine the range of checks for which a checking account at Long's Bank generates more earnings than at Fellow's Bank, we analyze the provided earnings equations: Long's Bank earnings are expressed as \( I_L = -0.06x + 8.3 \) and Fellow's Bank as \( I_F = -0.02x + 6.6 \). Setting up the inequality \( -0.06x + 8.3 > -0.02x + 6.6 \) allows us to isolate \( x \). Rearranging leads to \( 1.7 > 0.04x \), which simplifies to \( x < 42.5 \). Since \( x \) represents the number of checks written and must be a whole number, we conclude that Long's Bank will generate more earnings than Fellow's Bank when the number of checks written is in the range of \( x \in [0, 42] \). Therefore, for up to 42 checks, Long's Bank offers greater earnings potential.

Answer 1

To confirm the analysis, let's solve the inequality step-by-step.

You have the earnings equations for each bank:
- Long's Bank: \( I_L = -0.06x + 8.3 \)
- Fellow's Bank: \( I_F = -0.02x + 6.6 \)

To determine when Long's Bank generates more earnings than Fellow's Bank, we set up the inequality:

\[
-0.06x + 8.3 > -0.02x + 6.6
\]

Now, let's rearrange the inequality to isolate \( x \). Start by adding \( 0.06x \) and \( 6.6 \) to both sides:

\[
8.3 - 6.6 > -0.02x + 0.06x
\]

This simplifies to:

\[
1.7 > 0.04x
\]

Next, divide both sides by \( 0.04 \):

\[
\frac{1.7}{0.04} > x
\]

Calculating the left side gives:

\[
42.5 > x
\]

This simplifies to:

\[
x < 42.5
\]

Since \( x \) (the number of checks) must be a whole number, the possible integer values for \( x \) that satisfy the inequality are:

\[
x \in [0, 42]
\]

Thus, Long's Bank will generate greater earnings than Fellow's Bank for up to 42 checks written. Therefore, you are correct in concluding that for up to 42 checks, Long's Bank offers greater earnings potential.