Problem 64 - College Algebra

Question

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

The moving company becomes cheaper at 11 boxes, with a total cost of \( \$205 \).

Step 1: Define the Variables

Let's define the variables involved in this problem. Let \( x \) represent the number of boxes.

Step 2: Set Up the Equations

Set up the equations based on the problem statement. The total cost for the moving company is \( 150 + 5x \) dollars, where \( 150 \) is the flat rate and \( 5x \) is the cost per box. The total cost for the taxi service is \( 20x \) dollars, where \( 20 \) is the cost per box.

Step 3: Establish Inequality for Comparison

The goal is for the moving company to be cheaper than the taxi service. Thus, the inequality is: \[150 + 5x < 20x\]

Step 4: Solve the Inequality

Solve the inequality from Step 3. Begin by isolating \( x \) on one side:1. Rearrange the inequality: \[ 150 < 20x - 5x \] 2. Simplify:\[ 150 < 15x \] 3. Divide both sides by 15:\[ x > 10 \] This means for \( x = 11 \) boxes or more, the moving company is cheaper.

Step 5: Calculate the Total Cost

Now calculate the total cost for 11 boxes using the moving company:The total cost with the moving company is \( 150 + 5 \times 11 = 150 + 55 = 205 \) dollars. Therefore, with 11 boxes, the moving company charges \( \$205 \).

Key Concepts

Linear Inequality

Linear inequalities are mathematical expressions used to compare two quantities where one side is not necessarily equal to the other. Instead, the expression will use symbols like <, >, ≤, or ≥ to signify that one quantity is less than, greater than, less than or equal to, or greater than or equal to another. In our exercise, the linear inequality is used to determine when the moving company is cheaper than the taxi service by comparing their total costs.Here’s how it works:

• The total cost for the moving company is represented as \(150 + 5x\), where \(150\) is the flat rate and \(5x\) is the cost per box.
• The total cost for the taxi service is \(20x\), where each box incurs a charge of \(20\) dollars.
• We write the inequality as \(150 + 5x < 20x\) to express that the cost for the moving company should be less than the taxi service.

By solving this inequality, we're looking to find out how many boxes (\(x\)) would make the moving company's total cost cheaper than that of the taxi service. By analyzing signs and operations in such inequalities, we can set conditions and constraints to solve real-world problems efficiently.

Cost Comparison

Cost comparison is key in making financial decisions, just like in everyday situations where you decide between two products or services based on their prices. In this problem, we compare the costs of the moving company and the taxi service to find out which is more affordable for shipping a certain number of boxes.In the scenario presented:

• Moving Company: Charges entail a flat fee of \(\\(150\) plus an additional \(\\)5\) per box.
• Taxi Service: Charges \(\$20\) per box, with no initial flat fee.

The decision revolves around finding the number of boxes where the total cost of the moving company becomes favorable. Solving the inequality \(150 + 5x < 20x\) tells us that when \(x > 10\), or 11 boxes or more, the moving company is the cheaper option.Such exercises enhance problem-solving skills by analyzing costs and making optimal decisions based on quantitative data. This understanding is crucial in various fields like budgeting, resource allocation, and economic analysis.

Variable Isolation

Variable isolation is a fundamental part of solving equations and inequalities. It involves manipulating the equation so that the variable of interest (often \(x\)) is alone on one side. This allows us to directly interpret its value or range. In our problem, we solved a linear inequality to determine when one option is cheaper than another.Here's how isolation works in this exercise:

• Starting with the inequality \(150 + 5x < 20x\), we aim to get \(x\) by itself on one side.
• We first consolidate terms by subtracting \(5x\) from both sides, giving us \(150 < 15x\).
• Next, we divide both sides by 15 to solve for \(x\), deriving \(x > 10\).

With \(x\) isolated, we conclude that more than 10 boxes (specifically 11 or more) will result in the moving company being the cheaper option.Isolating variables is a technique that transforms complex equations into a simplified form, making them easier to interpret and solve. It's an essential skill in algebra, providing a step-by-step approach to unravel mathematical relationships.