Solving Inequalities: Which Point Fits The Solution?

Hey everyone! Let's dive into a cool math problem together. We're gonna figure out which point from a list fits the inequality y ≥ 3x - 1. Sounds like fun, right? Basically, we're testing some coordinates to see which ones make the inequality true. This is super helpful for understanding how inequalities work on a graph. So, grab your pencils, and let's get started. This process is important because it is used for lots of different mathematical and real world applications. Understanding inequalities will allow you to do things like graph them on a coordinate plane, understanding how to apply constraints to real world problems. So let's get down to it!

Understanding the Inequality: y ≥ 3x - 1

Alright, before we start, let's break down what y ≥ 3x - 1 actually means. This inequality represents a region on a coordinate plane. The 'y ≥' part tells us that we're looking at all the points where the y-value is greater than or equal to the result of 3x - 1. If we were to graph this, it would be a line (3x - 1) and everything above that line. Points that satisfy the inequality will lie either on the line or in the region above it. If you're a visual learner, imagine a line on a graph. The inequality includes the line itself and everything above it. This means any point in that area is a solution. Also, the line is solid, that is, it includes the points on the line. It's like drawing a boundary and saying, 'Okay, everything on this side, including the line, is cool.' Any point on the line or in the shaded region is part of the solution set. Understanding this concept is really important, you are going to use it a lot as you move forward with more and more complex mathematical ideas. Also, you will see it in many different contexts. So, with this context let's find the correct answer!

Testing the Points

Now, let's get to the main event: testing the points! We've got four options, and we're going to plug each one into the inequality to see if it holds true. It's like a math detective game, and we're the detectives! We're going to substitute the x and y values of each point into y ≥ 3x - 1 and see if the inequality is valid. Each point is an (x, y) coordinate, so we just substitute those numbers into our equation and check our results. If it does hold true, then we know we've got a valid solution. Otherwise, we know it doesn't fit the inequality.

Point A: (-2.5, -9)

Let's start with point A, which is (-2.5, -9). We'll substitute x = -2.5 and y = -9 into our inequality: y ≥ 3x - 1. This gives us: -9 ≥ 3(-2.5) - 1*. Simplifying this, we get: -9 ≥ -7.5 - 1, which further simplifies to -9 ≥ -8.5. Now, is -9 greater than or equal to -8.5? Nope, it isn't. So, point A (-2.5, -9) does not satisfy the inequality.

Point B: (1, 1)

Next up, we have point B, which is (1, 1). Let's plug in x = 1 and y = 1: 1 ≥ 3(1) - 1*. This becomes: 1 ≥ 3 - 1, and simplifying further, we get: 1 ≥ 2. Is this true? No way! Therefore, point B (1, 1) does not fit the inequality either. Keep in mind that we're looking for y-values that are greater than or equal to the result of 3x-1. This is going to be important as we work through these examples. Since the x and y values are both positive in this case, we know that the y value needs to be higher in order to satisfy the inequality. Let's see if our next test works out.

Point C: (2, 5)

Okay, let's give point C (2, 5) a try. Substitute x = 2 and y = 5: 5 ≥ 3(2) - 1*. This simplifies to: 5 ≥ 6 - 1, which further simplifies to: 5 ≥ 5. Hey, is that true? Yep, it is! 5 is equal to 5. Since we have 'greater than or equal to,' this point does fit the inequality. So, point C (2, 5) does satisfy the inequality. The key here is to carefully evaluate the inequality. If you follow the rules of arithmetic, you should always get the correct answer. In this case, we have the correct answer and we are happy about it.

Point D: (-1, -5)

Finally, let's check out point D, which is (-1, -5). Plugging in x = -1 and y = -5, we get: -5 ≥ 3(-1) - 1*. This becomes: -5 ≥ -3 - 1, which simplifies to: -5 ≥ -4. Is this true? Nope, it's not. So, point D (-1, -5) does not satisfy the inequality. -5 is less than -4. This does not satisfy the inequality. And that's all of the points! Now we have our answer.

Conclusion: The Solution Set

Alright, we tested all the points, and we found that point C (2, 5) is the only one that satisfies the inequality y ≥ 3x - 1. This means that if we were to graph this inequality, the point (2, 5) would either lie on the line y = 3x - 1 or be in the region above it. So, the correct answer is option C. Isn't that neat? By checking each point, we've figured out which one fits the solution set. Remember, understanding how to test these points is a useful skill. This skill goes beyond this specific problem. You can apply it to many other mathematical and real-world problems. Keep practicing, and you'll become a pro at solving inequalities! Also, remember the general concept of what you are working with. We are working with an inequality. Understanding the difference between an inequality and an equation is key to understanding this. If you are ever confused, then you can go back to the basic definitions.