Solving Systems Of Linear Inequalities: Ordered Pair Solutions

Hey guys! Let's dive into the world of linear inequalities and ordered pairs. We're going to figure out how to determine which ordered pairs are solutions to a system of linear inequalities. It might sound a bit complex, but trust me, we'll break it down step by step. So, grab your thinking caps, and let's get started!

Understanding Linear Inequalities

Before we jump into solving, let's make sure we're all on the same page about what linear inequalities actually are. Linear inequalities are mathematical expressions that compare two values using inequality symbols like >, <, ≥, or ≤. Unlike linear equations, which have a single solution (or sometimes no solution), linear inequalities have a range of solutions. When we're dealing with a system of linear inequalities, we're looking for the set of ordered pairs that satisfy all the inequalities in the system.

Think of it like this: imagine you have two rules you need to follow at the same time. A solution to the system is any point that obeys both rules simultaneously. Graphically, each inequality represents a region on the coordinate plane, and the solution set is the overlapping region where all inequalities are true. This overlapping region is often called the feasible region. To identify solutions, we need to understand how to graph these inequalities and then test ordered pairs to see if they fall within the feasible region.

Key Concepts of Linear Inequalities

First, let’s define what a linear inequality is. A linear inequality is similar to a linear equation, but instead of an equals sign (=), it has an inequality sign (>, <, ≥, ≤). For example, y > 2x + 1 or y ≤ -x + 3 are linear inequalities. These inequalities represent a region on the coordinate plane, not just a single line. The solutions to a linear inequality are all the points (ordered pairs) that make the inequality true.

When we deal with a system of linear inequalities, we’re looking for the set of ordered pairs that satisfy all the inequalities in the system simultaneously. This means the ordered pair must make each inequality true. For instance, if we have two inequalities, an ordered pair is a solution only if it works for both inequalities.

Graphically, a linear inequality represents a half-plane. The boundary line (e.g., y = 2x + 1) divides the coordinate plane into two regions. For inequalities with > or <, the boundary line is dashed to indicate that points on the line are not included in the solution. For inequalities with ≥ or ≤, the boundary line is solid, meaning points on the line are included in the solution. The solution to the inequality is one of these half-planes, and we typically shade the region that represents the solution.

Solving Systems Graphically

To solve a system of linear inequalities graphically, you graph each inequality on the same coordinate plane. The region where the shaded areas of all inequalities overlap is the solution set. Any point in this overlapping region is a solution to the system. This method is particularly useful because it provides a visual representation of the solution set.

For example, consider the system:

y > x + 1
y < -x + 3

First, graph each inequality. The solution to y > x + 1 is the region above the dashed line y = x + 1. The solution to y < -x + 3 is the region below the dashed line y = -x + 3. The area where these two regions overlap is the solution to the system. Any point in this overlapping region satisfies both inequalities.

The Problem: Finding Ordered Pair Solutions

Now, let's tackle the specific problem we're faced with. We have a system of two linear inequalities:

y ≥ -½x
y < ½x + 1

And we need to determine which of the given ordered pairs, namely (5, -2), (3, 1), and (4, 2), are solutions to this system. Remember, an ordered pair is a solution if and only if it satisfies both inequalities. This means when you plug the x and y values from the ordered pair into each inequality, the resulting statements must be true.

Step-by-Step: Testing Ordered Pairs

So, how do we do this? We'll take each ordered pair one at a time and substitute the x and y values into both inequalities. If both inequalities hold true, then that ordered pair is a solution. If even one inequality is false, then the ordered pair is not a solution.

Let’s start with the first ordered pair, (5, -2). We'll plug x = 5 and y = -2 into our inequalities:

• For the first inequality, y ≥ -½x, we get:

  -2 ≥ -½(5)
  -2 ≥ -2.5
  

  Is this true? Yes, -2 is indeed greater than -2.5. So, the first inequality holds for (5, -2).

• Now, let's check the second inequality, y < ½x + 1:

  -2 < ½(5) + 1
  -2 < 2.5 + 1
  -2 < 3.5
  

  Is this true? Yes, -2 is less than 3.5. So, the second inequality also holds for (5, -2).

Since both inequalities are true for (5, -2), this ordered pair is a solution to the system.

Now, let's move on to the second ordered pair, (3, 1). We'll repeat the process:

• For the first inequality, y ≥ -½x:

  1 ≥ -½(3)
  1 ≥ -1.5
  

  This is true, since 1 is greater than -1.5.

• For the second inequality, y < ½x + 1:

  1 < ½(3) + 1
  1 < 1.5 + 1
  1 < 2.5
  

  This is also true, since 1 is less than 2.5.

Therefore, (3, 1) is a solution to the system.

Finally, let's test the third ordered pair, (4, 2):

• For the first inequality, y ≥ -½x:

  2 ≥ -½(4)
  2 ≥ -2
  

  This is true, since 2 is greater than or equal to -2.

• For the second inequality, y < ½x + 1:

  2 < ½(4) + 1
  2 < 2 + 1
  2 < 3
  

  This is also true, since 2 is less than 3.

So, (4, 2) is a solution to the system as well.

Applying the Solution

We've methodically tested each ordered pair and found that (5, -2), (3, 1), and (4, 2) all satisfy both inequalities in the system. This means all three ordered pairs are solutions to the system of linear inequalities.

Practical Application: Why This Matters

You might be wondering,