Find Ordered Pairs For Inequalities

Hey math whizzes! Ever stared at a problem and thought, "Which ordered pair actually works here?" You're not alone, guys! Today, we're diving deep into the awesome world of inequalities and figuring out how to pinpoint the exact ordered pair that makes both of them happy. It sounds tricky, but trust me, with a little practice and some killer strategies, you'll be a pro in no time. We're talking about cracking the code on systems of inequalities, and it's more fun than you think! Get ready to flex those math muscles because we're about to break down a classic problem that will have you understanding these concepts like never before. So, buckle up, grab your favorite calculator (or just your amazing brainpower), and let's get this math party started!

Understanding the Problem: What's an Ordered Pair Anyway?

Alright, let's get down to brass tacks. The question asks: Which ordered pair makes both inequalities true? We've got two inequalities: y < 3x - 1 and y >= -x + 4. An ordered pair, which looks like (x, y), is just a fancy way of saying a point on a graph. It has an x-coordinate and a y-coordinate. For an ordered pair to be a solution to a system of inequalities, it has to satisfy every single inequality in that system. Think of it like a lock and key; the ordered pair has to fit both locks (the inequalities) perfectly. If it only fits one, it's a no-go. We're given four potential keys: A. (4,0), B. (1,2), C. (0,4), and D. (2,11). Our mission, should we choose to accept it, is to test each of these pairs in both inequalities. If an ordered pair works in both, bingo, that's our answer! If it fails even one, we toss it aside and move to the next contender. This process is fundamental to understanding how to graph and interpret solutions to systems of inequalities, which is a cornerstone of algebra. The visual representation of these inequalities on a coordinate plane helps us understand the concept of a solution set as a region rather than just discrete points. The boundary lines of these inequalities are either solid (for <= or >=) or dashed (for < or >), indicating whether points on the line are included in the solution set. The shading of the regions above or below these lines represents all the possible y-values for a given x-value that satisfy the inequality. When we have two inequalities, the solution to the system is the overlap of the shaded regions, representing the points that satisfy both conditions simultaneously. Our task is to find a specific point within this overlap.

Strategy Session: Testing Our Ordered Pairs

So, how do we actually do this? The most straightforward method is substitution. We take each ordered pair (x, y) and plug its x and y values into both inequalities. Let's break down the inequalities first. The first one is y < 3x - 1. This means the y-value must be strictly less than the result of 3x - 1. The second one is y >= -x + 4. Here, the y-value must be greater than or equal to the result of -x + 4. It's crucial to pay attention to the inequality signs: < (less than) means the line is dashed and points on the line don't count, while >= (greater than or equal to) means the line is solid and points on the line do count. Now, let's get our hands dirty with the test cases. We'll start with option A, the ordered pair (4,0). Here, x = 4 and y = 0. We plug these into the first inequality: 0 < 3(4) - 1. This simplifies to 0 < 12 - 1, which is 0 < 11. Is this true? Yep, 0 is indeed less than 11. Great start! Now, we must test it in the second inequality: 0 >= -4 + 4. This simplifies to 0 >= 0. Is this true? Yes, 0 is equal to 0, so it satisfies the condition. Since (4,0) makes both inequalities true, it's a potential answer! However, to be absolutely sure and to solidify our understanding, we should continue testing the other options. This rigorous approach prevents errors and builds confidence in our mathematical reasoning. Remember, in mathematics, proof and verification are key. Just because one option seems right doesn't mean we should stop; diligent checking ensures accuracy. This systematic substitution method is a universal technique applicable to any system of linear equations or inequalities, providing a reliable way to verify solutions. It’s like being a detective, gathering evidence (the results of the substitutions) to solve the case. Each test brings us closer to the definitive answer, reinforcing the logical progression of mathematical problem-solving. So, let's keep going and see what the other pairs reveal!

Testing the Options: A Step-by-Step Breakdown

Let's dive into the nitty-gritty of testing each ordered pair. This is where the magic happens, guys!

Option A: (4,0)

We already did a quick check, but let's formalize it. For (4,0), x = 4 and y = 0.

• Inequality 1: y < 3x - 1

  • Substitute: 0 < 3(4) - 1
  • Simplify: 0 < 12 - 1
  • Result: 0 < 11. This is TRUE.

• Inequality 2: y >= -x + 4

  • Substitute: 0 >= -(4) + 4
  • Simplify: 0 >= -4 + 4
  • Result: 0 >= 0. This is TRUE.

Since both inequalities are true for (4,0), this is a very strong candidate. But let's not stop here!

Option B: (1,2)

For (1,2), x = 1 and y = 2.

• Inequality 1: y < 3x - 1
  • Substitute: 2 < 3(1) - 1
  • Simplify: 2 < 3 - 1
  • Result: 2 < 2. This is FALSE. (2 is not strictly less than 2).

Since this ordered pair fails the first inequality, we don't even need to check the second one. It's automatically disqualified. Ouch!

Option C: (0,4)

For (0,4), x = 0 and y = 4.

• Inequality 1: y < 3x - 1
  • Substitute: 4 < 3(0) - 1
  • Simplify: 4 < 0 - 1
  • Result: 4 < -1. This is FALSE.

Again, this pair fails the first inequality, so it's out. Better luck next time!

Option D: (2,11)

For (2,11), x = 2 and y = 11.

• Inequality 1: y < 3x - 1
  • Substitute: 11 < 3(2) - 1
  • Simplify: 11 < 6 - 1
  • Result: 11 < 5. This is FALSE.

Wow, option D also fails the first inequality. It seems our initial candidate is looking really good!

The Verdict: Which Pair Wins?

After meticulously testing each ordered pair, we found that only (4,0) satisfied both inequalities: y < 3x - 1 and y >= -x + 4. Let's recap why the others didn't make the cut:

• (1,2) failed y < 3x - 1 because 2 is not less than 2.
• (0,4) failed y < 3x - 1 because 4 is not less than -1.
• (2,11) failed y < 3x - 1 because 11 is not less than 5.

This systematic approach is key to mastering systems of inequalities. Remember, for an ordered pair to be a solution, it must satisfy all conditions. The process involves careful substitution and checking the inequality signs. This method is super reliable, and once you get the hang of it, you'll be solving these problems in a flash. It’s not just about getting the right answer; it’s about understanding why it’s the right answer. Visualizing these inequalities on a graph can also be incredibly helpful. The first inequality y < 3x - 1 would be a dashed line with a slope of 3 and a y-intercept of -1, with the region below the line shaded. The second inequality y >= -x + 4 would be a solid line with a slope of -1 and a y-intercept of 4, with the region above the line shaded. The solution to the system is the area where these shaded regions overlap. Our winning point, (4,0), lies within this overlapping region, confirming its validity. Understanding this graphical interpretation adds another layer to your comprehension and provides a visual check for your algebraic solutions. Keep practicing, keep exploring, and you'll become an inequality ninja in no time! You guys are doing great!

Conclusion: Mastering Inequalities

So there you have it, math adventurers! We've successfully navigated the world of inequalities and found the single ordered pair that makes both equations sing in harmony. The key takeaway here is the power of substitution and verification. Don't just guess; test! Plug those x and y values into every inequality provided. Pay close attention to those greater than, less than, and equal to signs – they matter! This methodical approach isn't just for this specific problem; it's a fundamental skill that will serve you well in all areas of mathematics. Whether you're dealing with simple linear inequalities or complex systems, the principle remains the same: prove it works for all conditions. The more you practice, the more intuitive this process becomes. You'll start to recognize patterns and develop a feel for which solutions are likely to be correct. Remember, math is like a muscle; the more you work it out, the stronger it gets. So, keep tackling those problems, ask questions, and don't be afraid to make mistakes – they're just stepping stones on your learning journey. Keep up the fantastic work, and remember to always check your answers! You've got this!