In pre-calculus class, we learn about systems of inequalities. These are important tools that help us find the best answers when there are limits or constraints. We use them in different areas like economics (money matters), engineering (designing things), and logistics (how to move goods).

What are Systems of Inequalities?

A system of inequalities is when we have two or more inequalities that we solve together. Let’s look at some examples:

1. (2x + 3y \leq 12)
2. (x - y \geq 2)
3. (x \geq 0)
4. (y \geq 0)

These inequalities create a specific area on a graph. This area is called the feasible region, and it shows where all the conditions are met.

How to Find the Best Solutions

When solving an optimization problem, we want to either make something as big as possible (like profit) or as small as possible (like cost). We often represent this goal with a formula like (z = ax + by).

For example, let’s say we want to maximize profit, which we can write as (z = 3x + 4y). Here’s how we find the best solution:

1. Graph the inequalities: Draw each inequality on a graph.
2. Find the feasible region: Look for the overlapping area where all inequalities are true. This is our feasible region.
3. Evaluate the objective function: Check the value of (z) at the corners of the feasible region. The best solutions usually happen at these points.

An Example

Let’s take one of the points from our graph, like ((2, 2)), and plug it into our profit equation:

[z = 3(2) + 4(2) = 6 + 8 = 14]

By looking at all the points, you can find the highest value, which will help you make the best decision.

In short, systems of inequalities help us strengthen our problem-solving skills and give us the tools we need to solve real-life optimization problems.