Sketch The Solution To The System Of Inequalities

Sketch the Solution to the System of Inequalities: A Step-by-Step Guide

Introduction

Imagine you’re a small business owner trying to maximize profits while staying within budget. You have two products: Product A and Product B. Product A requires 2 hours of labor and $5 in materials per unit, while Product B requires 3 hours of labor and $4 in materials per unit. Your total labor hours available are 60 per week, and your material budget is $100. How do you determine the optimal number of each product to produce?

This scenario is a classic example of a system of inequalities—a set of mathematical constraints that define the limits of a problem. Systems of inequalities are foundational in fields like economics, engineering, and operations research. In this article, we’ll explore how to sketch the solution to a system of inequalities, step by step, and understand why this skill is critical for solving real-world problems.

What Is a System of Inequalities?

A system of inequalities consists of two or more inequalities that share the same variables. The solution to the system is the set of all points that satisfy all the inequalities simultaneously. Graphically, this solution is represented as the intersection of the regions defined by each inequality.

For example, consider the system:

1. $ x + y \leq 5 $
2. $ 2x - y \geq 3 $

Each inequality defines a region on the coordinate plane. The solution to the system is the area where these regions overlap.

Why Graphing Systems of Inequalities Matters

Graphing systems of inequalities is a powerful tool for visualizing constraints and identifying feasible solutions. It allows you to:

• Identify trade-offs: Understand how changing one variable affects others.
• Optimize resources: Find the best possible outcome within given limits.
• Solve real-world problems: From budgeting to logistics, systems of inequalities model practical scenarios.

Let’s break down the process of sketching the solution to a system of inequalities.

Step-by-Step Guide to Sketching the Solution

Step 1: Graph Each Inequality Individually

Start by treating each inequality as a linear equation. For example, take $ x + y \leq 5 $.

• Rewrite the inequality as an equation: $ x + y = 5 $.
• Find the intercepts:
  • When $ x = 0 $, $ y = 5 $.
  • When $ y = 0 $, $ x = 5 $.
• Plot the line: Draw a straight line through the points (0, 5) and (5, 0).
• Shade the region: Since the inequality is $ \leq $, shade the area below the line (including the line itself).

Repeat this process for the second inequality, $ 2x - y \geq 3 $.

• Rewrite as an equation: $ 2x - y = 3 $.
• Find intercepts:
  • When $ x = 0 $, $ y = -3 $.
  • When $ y = 0 $, $ x = 1.5 $.
• Plot the line: Draw a line through (0, -3) and (1.5, 0).
• Shade the region: Since the inequality is $ \geq $, shade the area above the line (including the line).

Step 2: Identify the Intersection of the Shaded Regions

The solution to the system is the area where the shaded regions of both inequalities overlap. In our example, this is the region that lies below $ x + y = 5 $ and above $ 2x - y = 3 $.

Step 3: Test a Point to Confirm the Correct Region

To ensure you’ve shaded the correct side of each line, pick a test point (e.g., the origin (0, 0)) and substitute it into the inequalities:

• For $ x + y \leq 5 $: $ 0 + 0 = 0 \leq 5 $ → True.
• For $ 2x - y \geq 3 $: $ 2(0) - 0 = 0 \geq 3 $ → False.

Since the origin does not satisfy the second inequality, the correct region is the opposite side of the line $ 2x - y = 3 $.

Step 4: Label the Solution Region

Clearly mark the overlapping area as the solution to the system. This region represents all possible combinations of $ x $ and $ y $ that satisfy both constraints.

Real-World Example: Optimizing Production

Let’s revisit the business example. Suppose the company wants to maximize profit, with Product A generating $10 per unit and Product B generating $12 per unit. The profit function is $ P = 10x + 12y $.

By graphing the system of inequalities:

1. $ 2x + 3y \leq 60 $ (labor constraint)
2. $ 5x + 4y \leq 100 $ (material constraint)
3. $ x \geq 0 $, $ y \geq 0 $ (non-negativity)

The overlapping region defines the feasible production options. The optimal solution lies at

the corner point of this feasible region that maximizes the profit function. To find this point, we evaluate the profit function at each corner: (0,0), (0,15), (20,0), and the intersection of the lines $2x + 3y = 60$ and $5x + 4y = 100$.

Solving the system of equations: $2x + 3y = 60$ $5x + 4y = 100$

Multiply the first equation by 5 and the second by 2: $10x + 15y = 300$ $10x + 8y = 200$

Subtract the second equation from the first: $7y = 100$ $y = \frac{100}{7} \approx 14.29$

Substitute this value of $y$ back into the first equation: $2x + 3(\frac{100}{7}) = 60$ $2x + \frac{300}{7} = 60$ $2x = 60 - \frac{300}{7} = \frac{420 - 300}{7} = \frac{120}{7}$ $x = \frac{60}{7} \approx 8.57$

So the intersection point is $(\frac{60}{7}, \frac{100}{7})$. Now let's calculate the profit at each corner:

• (0,0): $P = 10(0) + 12(0) = 0$
• (0,15): $P = 10(0) + 12(15) = 180$
• (20,0): $P = 10(20) + 12(0) = 200$
• $(\frac{60}{7}, \frac{100}{7})$: $P = 10(\frac{60}{7}) + 12(\frac{100}{7}) = \frac{600}{7} + \frac{1200}{7} = \frac{1800}{7} \approx 257.14$

The maximum profit occurs at the point $(\frac{60}{7}, \frac{100}{7})$. Therefore, the company should produce $\frac{60}{7}$ units of Product A and $\frac{100}{7}$ units of Product B to maximize their profit, resulting in a maximum profit of approximately $257.14.

Conclusion

Sketching the solution to a system of inequalities is a powerful visual tool for understanding the feasible region and identifying the possible combinations of variables that satisfy the given constraints. By systematically graphing each inequality, finding the intersection of the shaded regions, and testing a point, we can accurately determine the solution set. This technique is not only valuable in mathematical contexts but also has practical applications in various fields, including business optimization, resource allocation, and engineering design, enabling informed decision-making based on defined limitations. The ability to visualize and analyze these constraints provides a clearer understanding of the potential outcomes and allows for the identification of optimal solutions.