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What is a Real-Life Example of a System of Inequalities?

Published in Systems of Inequalities 2 mins read

A real-life example of a system of inequalities is a shop trying to buy stock with a given budget. This scenario perfectly illustrates how multiple constraints can be represented using inequalities.

Understanding the Scenario

Imagine a small shop owner who wants to purchase both shirts and pants for their store. They have a limited budget, say $1000, and each shirt costs $10 while each pair of pants costs $25. The shop owner wants to buy at least 20 shirts and at least 10 pairs of pants. These conditions create a system of inequalities.

Defining the Inequalities

Let's use variables:

  • x represents the number of shirts.
  • y represents the number of pants.

The inequalities representing the constraints are:

  • Budget constraint: 10x + 25y ≤ 1000 (The total cost must be less than or equal to $1000)
  • Minimum shirt requirement: x ≥ 20 (At least 20 shirts must be purchased)
  • Minimum pant requirement: y ≥ 10 (At least 10 pairs of pants must be purchased)

This system of three inequalities defines the feasible region—the set of all possible combinations of shirts and pants the shop owner can buy while meeting all the requirements. Solving this system graphically will show the feasible region, which would then help the shop owner make informed decisions about their stock.

Other Examples

Many other real-world problems can be modeled using systems of inequalities:

  • Dietary planning: Balancing calorie intake, macronutrient ratios (proteins, carbohydrates, fats), and budget constraints.
  • Manufacturing: Meeting production quotas while staying within material and labor limitations.
  • Resource allocation: Distributing limited resources (e.g., time, money, manpower) amongst multiple projects.
  • Financial planning: Managing expenses and savings while considering various income streams and debt levels.

The CK-12 Foundation and Nagwa websites provide further examples and explanations of how linear inequalities can be applied to real-world scenarios. As stated by Khan Academy, these inequalities are "used to model constraints for a manufacturing company or to create a budget in personal finance."

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