Checking 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)

Question

GPT-4o mini · Answer

To determine for what range of checks $ x $ the earnings at Long's Bank will be greater than the earnings at Fellow's Bank, we need to set up the inequalities based on the given equations. The earnings equation for Long's Bank is: $$ I_L = -0.06x + 8.3 $$ The earnings equation for Fellow’s Bank is: $$ I_F = -0.02x + 6.6 $$ We want to find when Long's Bank earnings are greater than Fellow's Bank earnings: $$ I_L > I_F $$ Substituting the equations into the inequality: $$ -0.06x + 8.3 > -0.02x + 6.6 $$ Now, we can solve this inequality step by step. First, let's isolate the variable $ x $. 1. Add $ 0.06x $ to both sides: $$ 8.3 > 0.04x + 6.6 $$ 2. Subtract $ 6.6 $ from both sides: $$ 8.3 - 6.6 > 0.04x $$ $$ 1.7 > 0.04x $$ 3. Divide both sides by $ 0.04 $: $$ \frac{1.7}{0.04} > x $$ Calculating $ \frac{1.7}{0.04} $: $$ 42.5 > x $$ or $$ x < 42.5 $$ This part tells us that for values of $ x $ less than 42.5, Long's Bank generates more income than Fellow's Bank. 4. Next, we need to determine the lower limit for $ x $. Since the number of checks written can't be negative in practical terms, we consider $ x \geq 0 $. Putting this together, the range of $ x $ where Long's Bank generates more income than Fellow's Bank is: $$ 0 \leq x < 42.5 $$ Thus, the final answer is that Long's Bank will generate more earnings than Fellow's Bank for: $$ \text{Range of } x: [0, 42.5) $$