Calculating Profit with Inequalities
Calculating profit with inequalities means writing an inequality where the revenue minus cost must be greater than zero (or some target) and solving for the unknown quantity. For example, if a company earns 12 dollars per unit and has fixed costs of 200 dollars, profit is positive when 12x - 200 > 0, which gives x > 16.67, so at least 17 units must be sold. Chapter 3 of OpenStax Elementary Algebra 2E presents this as a real-world application of linear inequalities. This type of problem forms the basis of break-even analysis used in every business and economics context.
Key Concepts
Property Profit is the money that remains when the expenses have been subtracted from the money earned. To find the number of jobs or sales needed to make a certain amount of profit, use the inequality: Revenue Expenses $\geq$ Target Profit.
Examples A landscaper has monthly expenses of 1,500 dollars. If he charges 75 dollars per job, how many jobs must he do to earn a profit of at least 3,000 dollars? Let $j$ be jobs. $75j 1500 \geq 3000$ solves to $j \geq 60$. He must do at least 60 jobs.
A jewelry maker sells necklaces for 50 dollars each. Her monthly expenses are 800 dollars. How many necklaces must she sell to make a profit of at least 1,000 dollars? Let $n$ be necklaces. $50n 800 \geq 1000$ solves to $n \geq 36$. She must sell at least 36 necklaces.
Common Questions
How do you calculate profit using inequalities?
Write profit as Revenue minus Cost, set up an inequality based on your goal (e.g., profit > 0 for break-even), then solve the inequality for the unknown quantity.
What is a break-even point?
The break-even point is where revenue equals cost, giving zero profit. To find it, solve Revenue = Cost. To make a profit, Revenue must exceed Cost, so Revenue > Cost.
What is a linear profit inequality?
A linear profit inequality has the form px - c > 0, where p is the price per unit, x is the number of units, and c is the total cost. Solving gives the minimum units needed for profit.
How do I solve a profit inequality?
Isolate the variable by applying the same inverse operations as solving an equation. Remember: multiplying or dividing both sides by a negative number reverses the inequality sign.
When do students learn profit applications with inequalities?
This is an algebra 1 application covered in OpenStax Elementary Algebra 2E Chapter 3: Math Models.
What is a common mistake in profit inequality problems?
Forgetting to flip the inequality sign when dividing by a negative number, which gives the wrong direction for the solution.
How does this relate to real-world business?
Every business uses break-even analysis. Profit inequalities model when a product becomes profitable, making this one of the most directly applicable algebra skills.