Problem 326 - Elementary Algebra

Question

In the following exercises, solve. Noe installs and configures software on home computers. He charges \(\$ 125\) per job. His monthly expenses are \(\$ 1,600\). How many jobs must he work in order to make a profit of at least \(\$ 2,400 ?\)

Step-by-Step Solution

Answer

Noe needs to complete at least 32 jobs.

Step 1: Define the Variables

Let\( \ x \) be the number of jobs Noe must work in a month. He charges \( \$125 \) per job. His monthly expenses are \( \$1600 \). To make a profit of at least \( \$2400 \), his total income must be his expenses plus his desired profit.

Step 2: Set Up the Equation

Noe's total income from \( \ x \) jobs is \( 125x \). He wants his income to be at least equal to his expenses plus his desired profit. Therefore, set up the inequality: \( 125x \geq 1600+2400 \)

Step 3: Simplify the Inequality

Combine the expenses and desired profit on the right-hand side of the inequality: \( 1600 + 2400 = 4000 \). So, the inequality simplifies to \( 125x \geq 4000 \)

Step 4: Solve for \( x \)

Divide both sides of the inequality by 125 to isolate \( x \): \( x \geq \frac{4000}{125} \). Simplifying the division, you get: \( x \geq 32 \)

Step 5: Interpret the Results

Noe needs to complete at least 32 jobs in a month to make a profit of at least \( \$2400 \).

Key Concepts

Linear Inequality

A linear inequality is similar to a linear equation but uses inequality symbols like \(< \), \(> \), \(eq \), \(≤ \), and \(≥ \) instead of an equal sign. In this exercise, we used the inequality \(125x \geq 4000\) to represent Noe's income condition.
Linear inequalities generally involve:

• Defining variables
• Constructing the inequality based on the problem
• Simplifying the inequality
• Solving the simplified inequality

The concept helps you understand constraints or limits within which a particular scenario, like profit, can occur. It allows us to visualize the range of possible solutions rather than a single value.

Profit Calculation

Profit is calculated by subtracting expenses from income. For Noe, his profit goal was at least \(\text{\$2400}\).
In mathematical terms, \( \text{Profit} = \text{Income} - \text{Expenses} \).
To determine the minimal number of jobs required, we added the profit goal to the expenses in our inequality. Thus the equation becomes an inequality representing the minimum necessary condition for profit:

• Noe’s income per job: \(\text{\$125} \)
• Monthly expenses: \(\text{\$1600} \)
• Desired profit: \(\text{\$2400}\)
• Total amount to cover: \(\text{\$1600} + \text{\$2400} = \text{\$4000} \)

Setting up this inequality: \(125x \geq 4000 \) allowed us to solve for \(\text{x}\), the number of jobs needed.

Variable Definition

In any mathematical problem, defining variables is crucial. They represent unknown quantities that you need to find. In this example, we defined:
Noe's number of jobs in a month as \(x\).
The income per job as \(125x\).
This step ensures clarity. You know exactly what each term in your inequality or equation represents. It also simplifies constructing and solving the inequality. Clear variable definitions also help in interpreting the results accurately.

Income and Expenses Analysis

Analyzing income and expenses helps to determine how many jobs you need to meet specific financial goals. For Noe:

• Income per job: \(\$125\)
• Monthly expenses: \(\$1600\)
• Desired profit: \(\$2400\)

Noe's total required income for profit can be calculated by combining his monthly expenses and desired profit. This total income must be at least \($4000 \) to make a profit of \(\$2400\).
Thus, we analyzed his income (from the jobs) and expenses to set up our linear inequality. We then solved it to find he needs at least 32 jobs to achieve his goal.