Solving Systems Of Inequalities: Find The Correct Ordered Pair

by Andrew McMorgan 63 views

Hey guys! Today, we're diving into the fascinating world of systems of inequalities and learning how to pinpoint the ordered pair that makes them both true. It might sound a bit intimidating at first, but trust me, it's like solving a puzzle, and we're going to crack it together. We'll break down the problem step-by-step, making sure everyone understands the logic behind each move. So, grab your thinking caps, and let's get started!

Understanding Systems of Inequalities

Before we jump into solving the specific problem, let's make sure we're all on the same page about what systems of inequalities actually are. Think of it like this: instead of having just one equation, we have two (or more!) inequalities that we need to satisfy simultaneously. Each inequality represents a region on a graph, and the solution to the system is the area where these regions overlap. An ordered pair is a solution to a system of inequalities if it makes all the inequalities true when you plug in the x and y values. This means that we are not just looking for one answer, but rather a set of answers that fit within the parameters of both inequalities. This concept is crucial in various real-world applications, such as optimizing resources, planning budgets, or even designing structures where multiple constraints must be met. Understanding how to solve these systems opens the door to a deeper comprehension of mathematical modeling and its practical uses. Remember, the key is to think of the inequalities as boundaries, and the solutions as the points that fall within the allowed regions defined by those boundaries. So, let’s keep this in mind as we move forward and tackle our example problem.

The Problem: Finding the Right Ordered Pair

Okay, let's tackle the problem at hand. We're given the following system of inequalities:

y < 3x - 1
y ≥ -x + 4

And we need to figure out which of the following ordered pairs satisfies both inequalities:

A. (4, 0)
B. (1, 2)
C. (0, 4)
D. (2, 1)

So, how do we go about this? The key here is to test each ordered pair one by one. We'll plug the x and y values from each pair into both inequalities and see if they hold true. If an ordered pair makes both inequalities true, then we've found our solution. If it fails even one of the inequalities, it's not the right answer. This methodical approach might seem a bit tedious, but it's the most reliable way to solve these types of problems. By carefully substituting the values and checking the results, we can systematically eliminate incorrect options and zero in on the correct one. This process not only helps us find the solution but also reinforces our understanding of how inequalities work and how they define regions on a graph. Remember, accuracy is crucial, so take your time, double-check your work, and let's find the ordered pair that fits the bill.

Testing the Ordered Pairs: A Step-by-Step Approach

Let's get our hands dirty and test each ordered pair! We'll go through each option (A, B, C, and D) systematically. For each pair, we'll substitute the x and y values into both inequalities and see if they hold true.

A. (4, 0)

  • Inequality 1: y < 3x - 1
    • Substitute: 0 < 3(4) - 1
    • Simplify: 0 < 12 - 1
    • Result: 0 < 11 (True)
  • Inequality 2: y ≥ -x + 4
    • Substitute: 0 ≥ -4 + 4
    • Simplify: 0 ≥ 0 (True)

Since (4, 0) makes both inequalities true, it looks like we might have a winner! But, to be absolutely sure, we need to test the other options as well.

B. (1, 2)

  • Inequality 1: y < 3x - 1
    • Substitute: 2 < 3(1) - 1
    • Simplify: 2 < 3 - 1
    • Result: 2 < 2 (False)

Since (1, 2) fails the first inequality, we can eliminate it right away. No need to test the second inequality.

C. (0, 4)

  • Inequality 1: y < 3x - 1
    • Substitute: 4 < 3(0) - 1
    • Simplify: 4 < 0 - 1
    • Result: 4 < -1 (False)

Again, (0, 4) fails the first inequality, so we can eliminate it.

D. (2, 1)

  • Inequality 1: y < 3x - 1
    • Substitute: 1 < 3(2) - 1
    • Simplify: 1 < 6 - 1
    • Result: 1 < 5 (True)
  • Inequality 2: y ≥ -x + 4
    • Substitute: 1 ≥ -2 + 4
    • Simplify: 1 ≥ 2 (False)

(2, 1) passes the first inequality but fails the second, so it's not the solution.

The Solution: Cracking the Code

Alright, guys, after meticulously testing each ordered pair, we've arrived at the solution! Remember, we were looking for the ordered pair that makes both inequalities true. Let's recap our findings:

  • (4, 0): Passed both inequalities.
  • (1, 2): Failed the first inequality.
  • (0, 4): Failed the first inequality.
  • (2, 1): Passed the first inequality but failed the second.

Therefore, the ordered pair that satisfies both inequalities is (4, 0). Option A is the correct answer! See? We cracked the code! This methodical approach of testing each option is a powerful tool when dealing with systems of inequalities. It ensures that you don't miss any potential solutions and helps you develop a deeper understanding of how inequalities work together. Remember, the key is to be patient, accurate, and to check your work carefully. Now you're one step closer to mastering the art of solving inequalities!

Visualizing the Solution: A Graphical Perspective

To really solidify our understanding, let's take a moment to visualize what we just did. Remember, each inequality represents a region on a graph. The solution to the system is the area where these regions overlap. If we were to graph the two inequalities, y < 3x - 1 and y ≥ -x + 4, we would see two distinct shaded regions.

The inequality y < 3x - 1 would be the area below the line y = 3x - 1 (and the line itself would be dashed because it's a strict inequality, meaning points on the line are not included). The inequality y ≥ -x + 4 would be the area above the line y = -x + 4 (and this line would be solid because the inequality includes the