Solving Inequalities: Finding Ordered Pair Solutions
Hey guys! Today, we're diving into the world of inequalities and ordered pairs. Specifically, we're tackling the question: Which ordered pairs are solutions to the inequality y - 3x < -4? This might sound a bit daunting at first, but trust me, it's totally manageable. We'll break it down step by step, and by the end, you'll be a pro at identifying solutions to inequalities. So, let's jump right in and get those brains working!
Understanding Inequalities and Ordered Pairs
Before we start crunching numbers, let's make sure we're all on the same page with the basics. An inequality, unlike a regular equation, doesn't have a single solution. Instead, it defines a range of possible values. Think of it as a boundary rather than a specific point. In our case, we have y - 3x < -4. This means we're looking for all the pairs of x and y values that, when plugged into the inequality, make the statement true.
An ordered pair, as you probably already know, is simply a pair of numbers written in a specific order, usually in the form (x, y). The first number represents the x-coordinate, and the second represents the y-coordinate. When we're dealing with inequalities in two variables (like x and y), ordered pairs represent points on a coordinate plane. Our goal is to figure out which of these points fall into the solution region defined by our inequality.
Why is this important? Well, inequalities pop up everywhere in real life. Imagine you're trying to figure out how many hours you can work to earn a certain amount of money, or how much of each ingredient you need for a recipe. Inequalities help us model these situations and find realistic solutions. Understanding how to solve them, and how ordered pairs fit into the picture, is a crucial skill in mathematics and beyond. So, let’s get down to business and see how we can find the solutions to our inequality!
The Method: Testing Ordered Pairs
The key to solving this type of problem is straightforward: we test each ordered pair in the given inequality. This involves substituting the x and y values from the ordered pair into the inequality and then simplifying to see if the resulting statement is true. If it's true, the ordered pair is a solution. If it's false, then it's not a solution. Simple as that!
Let's walk through this process step-by-step. Suppose we have an ordered pair (a, b). To test if it's a solution to y - 3x < -4, we replace x with 'a' and y with 'b' in the inequality. This gives us b - 3a < -4. Now, we simplify the left-hand side of the inequality. If the resulting value is less than -4, then the ordered pair (a, b) is indeed a solution. If it's greater than or equal to -4, then it's not a solution. We repeat this process for every ordered pair we're given.
The beauty of this method is its directness. There's no need to guess or estimate. We simply plug in the values and see what happens. However, it’s crucial to be careful with your arithmetic. A small mistake in your calculations can lead to a wrong conclusion. So, always double-check your work, and don't be afraid to use a calculator if needed. Now that we've got the method down, let's apply it to some specific ordered pairs and see how it works in practice.
Testing the Ordered Pairs: A Step-by-Step Guide
Alright, let's get our hands dirty and test some ordered pairs against our inequality y - 3x < -4. We'll go through each option one by one, showing you the calculations and the reasoning behind whether it's a solution or not. Remember, the goal is to substitute the x and y values into the inequality and see if the statement holds true.
1. Testing (-3, 0)
Our first ordered pair is (-3, 0). This means x = -3 and y = 0. Let's substitute these values into our inequality:
0 - 3(-3) < -4
Now, simplify:
0 + 9 < -4
9 < -4
Is this statement true? Nope! 9 is definitely not less than -4. So, the ordered pair (-3, 0) is not a solution to the inequality.
2. Testing (4, -2)
Next up is the ordered pair (4, -2), where x = 4 and y = -2. Let's plug these in:
-2 - 3(4) < -4
Simplify:
-2 - 12 < -4
-14 < -4
Is -14 less than -4? You bet! So, the ordered pair (4, -2) is a solution to our inequality.
3. Testing (0, -3)
Our third contender is the ordered pair (0, -3), with x = 0 and y = -3. Let's substitute:
-3 - 3(0) < -4
Simplify:
-3 - 0 < -4
-3 < -4
Is -3 less than -4? Nope, it's greater than -4. Therefore, (0, -3) is not a solution.
4. Testing (5, 1)
Now, let's try the ordered pair (5, 1), where x = 5 and y = 1:
1 - 3(5) < -4
Simplify:
1 - 15 < -4
-14 < -4
Again, -14 is less than -4, so (5, 1) is a solution to the inequality.
5. Testing (1, -1)
Finally, we have the ordered pair (1, -1), where x = 1 and y = -1. Let's plug these values in:
-1 - 3(1) < -4
Simplify:
-1 - 3 < -4
-4 < -4
Is -4 less than -4? No, it's equal to -4. Remember, our inequality is strictly less than -4, not less than or equal to. So, (1, -1) is not a solution.
The Solutions: Which Ordered Pairs Made the Cut?
Okay, guys, we've put each ordered pair through the wringer, and now it's time to gather our results. Remember, we were looking for the ordered pairs that, when plugged into the inequality y - 3x < -4, gave us a true statement. So, which ones made the cut?
Based on our calculations, the following ordered pairs are solutions to the inequality:
- (4, -2)
- (5, 1)
The other ordered pairs we tested, (-3, 0), (0, -3), and (1, -1), did not satisfy the inequality. They either resulted in a false statement or landed exactly on the boundary (in the case of (1, -1)).
So, there you have it! By systematically testing each ordered pair, we were able to identify the solutions to the inequality. This method is reliable and straightforward, making it a valuable tool in your mathematical arsenal. But let's think a bit more broadly about what these solutions actually represent.
Visualizing Solutions: The Graph of an Inequality
We've found the ordered pairs that satisfy our inequality algebraically, but it's also super helpful to visualize what's going on. The solutions to an inequality in two variables don't just exist as isolated points; they form a whole region on the coordinate plane. This region is the graph of the inequality.
Imagine drawing a line representing the equation y - 3x = -4. This line acts as the boundary for our inequality. But because we have a