Problem 125 - College Algebra

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A local bank charges \(\$ 8\) per month plus 5 e per check. The credit union charges \(\$ 2\) per month plus 8 d per check. How many checks should be written each month to make the credit union a better deal?

Step-by-Step Solution

Answer

To make the credit union a better deal than the bank, less than 2 checks should be written each month.

Step 1: Define the Variables and Set Up the Equations

Let \( c \) be the number of checks. Then, the total monthly charge for the bank is given by \( 8 + 5c \) and for the credit union, it is \( 2 + 8c \). In order to find the point at which the credit union becomes a better deal, we need to set up an inequality where the bank's charge is greater than the credit union's: \( 8 + 5c > 2 + 8c \).

Step 2: Simplify the Inequality

To simplify the inequality, subtract \( 5c \) from both sides to get: \( 8 > 2 + 3c \), then subtract 2 from both sides to get \( 6 > 3c \).

Step 3: Solve the Inequality for c

Divide both sides by 3 to solve for \( c \) in order to get \( c < 2 \).

Key Concepts

Understanding Word Problems

Word problems can often seem overwhelming at first because they require translating real-world scenarios into mathematical equations. However, the key to mastering word problems is to break them down into smaller, manageable parts. This exercise about bank and credit union charges is a great example.
First, identify what the problem is asking. Here, we need to determine how many checks should be written each month to make the credit union a better deal. This means comparing the costs from two different institutions and finding the threshold where one becomes more advantageous than the other.
Next, look at the details provided:

• Bank charges a constant monthly fee plus a per-check fee.
• Credit union charges a lower constant monthly fee but a higher per-check fee.

Finally, convert these verbal conditions into mathematical expressions, which leads us to set up inequalities for solving the problem.

Defining Variables in Word Problems

In any word problem, defining variables is fundamental to translating words into mathematical expressions. This involves identifying quantities that vary in the problem and assigning them symbols, usually letters.
For this exercise, let's consider the variable:

• Let \( c \) be the number of checks written each month.

Once defined, these variables help in setting up equations or inequalities that reflect the conditions of the problem. For instance, the bank's total monthly charge can be modeled as \( 8 + 5c \), which combines the set monthly cost and the cost per check.
The credit union, similarly, can be modeled as \( 2 + 8c \). With these expressions, you can compare costs directly by setting up an inequality.

Solving Linear Inequalities

When tackling linear inequalities, the goal is to find values of the variable that satisfy the inequality condition. This process is similar to solving linear equations, but with some additional rules, especially concerning the direction of the inequality sign.
Our inequality from the setup is: \[ 8 + 5c > 2 + 8c \]The steps to solve for \( c \) begin with simplifying both sides:

• Subtract \( 5c \) from both sides: \( 8 > 2 + 3c \)
• Subtract 2 from both sides: \( 6 > 3c \)
• Divide by 3 to isolate \( c \) : \( c < 2 \)

These steps illustrate manipulating the inequality to find the range of values for \( c \) that make the inequality true. Remember, when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips direction, though that's not needed in this particular problem. Finally, interpret this result in the context of the problem—writing fewer than 2 checks makes the credit union a better deal.