Problem 125 - Algebra and Trigonometry

Question

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A local bank charges 8 dollar per month plus 5 g per check. The credit union charges 2 dollar per month plus 8 g per check. How many checks should be written each month to make the credit union a better deal?

Step-by-Step Solution

Answer

The credit union is a better deal when more than 2 checks are written each month.

Step 1: Formulate the cost equations for both plans

Firstly, clearly express the costs associated with each plan. The bank's cost is given by \(B = 8 + 5g\), where \(g\) is the number of checks written. The credit union's cost is given by \(C = 2+ 8g\). These equations represent the total cost in dollars for the bank (\(B\)) and the credit union (\(C\)) respectively based on the number of checks written in a month.

Step 2: Set up the inequality

We are asked to find the number of checks where the credit union becomes cheaper than the bank. Therefore, we set up the inequality \(C < B\), plug in our previously defined cost equations, and get \(2+8g < 8+5g\). This inequality represents the cost of the credit union being less than the cost of the bank.

Step 3: Solve the inequality

Solve the inequality for \(g\): Subtract \(5g\) from both sides to get \(3g < 6\). Then divide both sides by 3 to get \(g > 2\). This solution indicates that when more than 2 checks are written in a month, it is cheaper to use the credit union.

Key Concepts

Linear Inequality Word Problems

Understanding linear inequality word problems involves breaking down the textual information into mathematical expressions. In real-life, such scenarios are common where we compare costs, rates, or quantities using inequalities.

In our exercise, we translate a situation comparing costs of two different plans into a linear inequality. This technique is essential for decision-making, like determining which plan offers a better financial option based on the usage level. The process starts by identifying the key variables and constants from the text — in this case, the number of checks ($g$) and the fixed monthly charges. We then represent the given information through linear equations before moving on to set up the inequality.

Steps to Tackling Word Problems

• Identify the unknown variable you need to solve for.
• Translate the text into a mathematical equation or inequality.
• Formulate linear expressions for the given conditions.
• Set up the inequality based on the scenario (e.g., one cost being less than the other).
• Solve the inequality to find the numerical answer.

When dealing with word problems, paying careful attention to words like 'more than', 'less than', 'at least', etc., helps in formulating the correct inequality and hence, leads to proper problem solving.

System of Linear Equations

A system of linear equations consists of two or more linear equations that have common solutions. When solving real-world problems, we often come across situations where multiple conditions must be satisfied simultaneously, which can be modeled through a system of linear equations.

However, in the case of our bank and credit union problem, we can see an overlap with a system of linear equations' concepts when initially setting up the separate cost equations for the bank and the credit union. They are not solved simultaneously like a classic system of linear equations but are instead compared. The inequation formed sets up a relationship that must be satisfied for one plan to be considered better than the other.

Such a relation surfaces often in budgeting, planning, and optimizing resources. Despite not fitting the traditional mold of simultaneous solutions, understanding how to dissect and create individual linear equations from complex word problems is an integral skill that underpins a broader range of mathematical problem-solving scenarios.

Inequality Problem Solving

Inequality problem solving distinguishes itself from regular equation solving by focusing on a range of possible solutions rather than a single solution. This makes it particularly relevant when dealing with problems that involve limitations, conditions, or thresholds, just like our bank charges problem.

Guidelines for Inequality Solving

To solve an inequality:

• Keep in mind the direction of the inequality. When multiplying or dividing both sides by a negative number, the inequality sign flips.
• Simplify the inequality just as you would do with an equation, by performing the same operations on both sides.
• When the inequality is solved, remember that the solution is often a range of values, not just a single number.

In our textbook problem, once the equations are set up, we get to the actual 'problem solving' aspect. The inequality symbol dictates the relationship between the expressions and represents a critical logical connector. We recognize that the credit union's cost must be less than the bank's for it to be the better deal (), and we process the inequality accordingly to solve for the number of checks ($g$). The result reveals a threshold beyond which one option becomes preferable over another.