Problem 64 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. A moving company charges a flat rate of $$\$ 150,$$ and an additional $$\$ 5$$ for each box. If a taxi service would charge $$\$ 20$$ for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

Step-by-Step Solution

Answer

For more than 10 boxes, the moving company is cheaper, with a cost of \$205 for 11 boxes.

Step 1: Define Variables

Let's define the variables. Let \( x \) represent the number of boxes. We have two costs to consider based on the number of boxes: the moving company's total cost and the taxi service's total cost.

Step 2: Formulate Equations

For the moving company, the cost \( C_m \) is given by a flat rate of \\(150 plus \\)5 per box: \( C_m = 150 + 5x \). For the taxi service, the cost \( C_t \) is \$20 per box: \( C_t = 20x \).

Step 3: Set Up Inequality

We need to find the number of boxes \( x \) for which the moving company's cost is less than the taxi service's cost. Thus, we need to solve the inequality: \( 150 + 5x < 20x \).

Step 4: Solve the Inequality

Starting with \( 150 + 5x < 20x \), subtract \( 5x \) from both sides to get \( 150 < 15x \). Then, divide both sides by 15 to isolate \( x \): \( x > 10 \).

Step 5: Calculate Total Cost for Comparison

When \( x = 11 \) (a value greater than 10), calculate the costs:- Moving company cost for 11 boxes: \( C_m = 150 + 5(11) = 205 \). - Taxi service cost for 11 boxes: \( C_t = 20(11) = 220 \). The moving company's cost is indeed cheaper for more than 10 boxes.

Key Concepts

Solving Inequalities

In the world of mathematics, solving inequalities is a common task akin to solving equations but with a few twists. An inequality shows a relationship where two expressions are not necessarily equal. Instead, they reveal one side is either greater than or less than the other.
Solving inequalities involves finding the values of a variable that make the inequality true. For example, with an expression like \( 150 + 5x < 20x \), our goal is to manipulate it just like an equation:

• Subtract \(5x\) from both sides to simplify the inequality.
• Divide by the coefficient of \(x\) to isolate \(x\).

This process helps in determining ranges of values instead of specific numbers, often leading us to solutions expressed as intervals or conditions, like \(x > 10\). These values are what make the inequality a true statement.

Linear Inequality

A linear inequality resembles a linear equation but instead of equality, it involves inequality symbols like <, >, ≤, and ≥. Linear inequalities are used to model situations where you need to consider ranges or conditions, rather than exact matches.
In our example, the inequality \(150 + 5x < 20x\) is linear due to its highest power of the variable being 1. This inequality sets a boundary where the cost by one company is less than another.

• The term \(150 + 5x\) represents a cost model involving a flat rate and an incremental cost per box.
• The expression \(20x\) is a simple proportional cost with no flat rate, signifying a variable-only cost model.

Linear inequalities are crucial for finding thresholds like when one option becomes cheaper than another, a typical use case in budgeting and financial planning.

Cost Comparison

Cost comparison is key in making smart financial decisions. It involves evaluating two or more cost models to determine which is more economical under certain conditions. In this exercise, we compared the costs of a moving company vs. a taxi service.
To compare:

• Understand each cost structure: a constant flat fee plus variable costs with the moving company, versus a strictly variable cost for the taxi.
• Use the inequality \(150 + 5x < 20x\) to find the threshold where one option becomes cheaper.

Calculating the number of boxes makes the moving company a better choice when \(x > 10\). Cost comparisons like these help you choose the best financial route, ensuring you save money by opting for services that offer better value under specific circumstances.

Variables in Equations

Variables are essential components in equations, serving as placeholders for unknown values that we aim to uncover. In solving real-world problems, they help bridge abstract mathematical concepts with tangible scenarios.
In our scenario, the variable \(x\) stands for the number of boxes.

• For the moving company, the equation \(C_m = 150 + 5x\) represents the total cost in terms of \(x\).
• The taxi service cost equation \(C_t = 20x\) directly links the total cost to the number of boxes, showing a one-to-one relationship between these entities.

Variables serve as a powerful tool in forming equations that help us model situations, analyze possibilities, and find optimal solutions in everyday decision-making tasks, like which service to use to minimize moving costs.