Problem 41 - Algebra 2

Question

MAIL For Exercises 40 and 41 , use the following information The U.S. Postal Service defines an oversized package as one for which the length \(L\) of its longest side plus the distance \(D\) around its thickest part is more than 108 inches and less than or equal to 130 inches. If the distance around the thickest part of a package= you want to mail is 24 inches, describe the range of lengths that would classify your package as oversized.

Step-by-Step Solution

Answer

The longest side must be more than 84 inches and less than or equal to 106 inches.

Step 1: Understand the Problem

We are given the condition that the sum of the length of the longest side \( L \) and the distance around the thickest part \( D \) of the package must be more than 108 inches and less than or equal to 130 inches. We know \( D = 24 \) inches. We need to find the possible values for \( L \).

Step 2: Set Up the Inequality

Using the problem's conditions, set up the inequality: \( L + 24 > 108 \) and \( L + 24 \leq 130 \). This follows because \( D = 24 \).

Step 3: Solve the First Inequality

Solve the inequality \( L + 24 > 108 \):\[ L + 24 > 108 \]Subtract 24 from both sides:\[ L > 84 \]

Step 4: Solve the Second Inequality

Solve the inequality \( L + 24 \leq 130 \):\[ L + 24 \leq 130 \]Subtract 24 from both sides:\[ L \leq 106 \]

Step 5: Combine the Results

Combine the solutions from steps 3 and 4. This gives the range of values for \( L \):\[ 84 < L \leq 106 \]

Step 6: Interpret the Result

The package is classified as oversized if its longest side \( L \) is greater than 84 inches and less than or equal to 106 inches.

Key Concepts

Linear Inequalities

Linear inequalities are an essential math concept used to express a range of values that a variable can take. Unlike linear equations, which show when two expressions are equal, inequalities show a less than or greater than relationship. For example, if you have an inequality like \( L + 24 > 108 \), this tells us that the sum must be greater than 108. Linear inequalities use symbols such as ">", "<", "≥", and "≤" to describe the relationships between values. These symbols help us calculate and express the possible solutions to mathematical problems involving inequities.

When solving linear inequalities, our goal is to isolate the variable. This allows us to determine all values the variable can logically take, given the problem constraints. The process is straightforward, involving similar steps to solving linear equations like adding, subtracting, multiplying, or dividing both sides by the same value. However, remember that if you multiply or divide an inequality by a negative value, the inequality sign flips. For instance, dividing by -1 would alter "<" to ">" and vice versa.

Understanding how to work with linear inequalities is crucial, as they often appear in real-world situations. They help define ranges for variables in problems involving constraints, such as dimensions in packaging or finance.

Range of Values

The range of values refers to the specific set of numbers that a variable can be when given a condition or inequality. In our postage problem, we are tasked with finding what length \(L\) can be, so the package remains oversized. Here, the range of values is represented as \( 84 < L \leq 106 \). This shows the lowest and highest values \(L\) can take to satisfy the inequalities we formed.

Here’s how you interpret this range:

• \(L\) must be greater than 84. This means 84 itself is not included, hence the "<" symbol.
• \(L\) can be 106 or any value less than that, hence the "\(\leq\)" symbol includes 106.

It's important to understand that range of values is indicative of all potential solutions that meet the problem criteria. Ranges can be open, closed, or half-open intervals and are an essential part of both mathematical problem-solving and real-life applications. They are foundational when dealing with anything that must sit within constraints, be it engineering specifications or environmental limits.

Problem Solving Steps

To efficiently solve problems like determining whether a package is oversized based on its dimensions, one can follow a structured set of problem-solving steps. These steps help ensure no important elements are overlooked and provide a clear understanding of how to arrive at an answer. Here is a breakdown of the steps used in our package problem:

• **Understand the Problem:** Start by comprehending what the problem is asking. Identify variables and constraints given, such as the sum condition and the fixed distance \(D = 24\) inches.
• **Set Up the Inequality:** Use the problem's conditions to construct inequalities involving the variable you need to find, which was \(L\). This involved setting equations like \( L + 24 > 108 \) and \( L + 24 \leq 130 \).
• **Solve the Inequalities:** Simplify each inequality to isolate \(L\). This involved subtracting \(24\) from each side to find \(L > 84\) and \(L \leq 106\).
• **Combine Results:** Take solutions from each inequality and combine them to find a solution set. This resulted in the range \( 84 < L \leq 106\).
• **Interpret Results:** Conclude what the range means for the problem. Understand that any \(L\) within this range classifies the package as oversized.

Following these steps not only helps solve the problem at hand but also builds a framework you can apply to other, similar challenges. It effectively translates mathematical theory into actionable processes.