Solving Inequalities: Find Ordered Pair Solutions For 2x + Y > -4
Let's dive into the world of inequalities and explore how to determine which ordered pairs satisfy a given inequality. In this case, we'll focus on the inequality 2x + y > -4. This type of problem is a fundamental concept in algebra and is crucial for understanding more advanced mathematical topics. We'll walk through the process step-by-step, making it clear and easy to follow. Our main goal is to identify which ordered pairs, from a given set of options, make the inequality true. So, grab your thinking caps, and let's get started!
Understanding Ordered Pairs and Inequalities
Before we jump into solving, let's make sure we're on the same page about what ordered pairs and inequalities are. An ordered pair, written as (x, y), represents a point on a coordinate plane. The first number, x, tells us how far to move horizontally from the origin (the point (0, 0)), and the second number, y, tells us how far to move vertically. For example, the ordered pair (2, 3) means we move 2 units to the right and 3 units up from the origin.
Now, what about inequalities? An inequality is a mathematical statement that compares two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). Unlike an equation, which states that two expressions are equal, an inequality indicates a range of possible values. In our case, we have the inequality 2x + y > -4, which means we're looking for ordered pairs (x, y) that, when plugged into the expression 2x + y, result in a value greater than -4.
So, to solve our problem, we'll take each ordered pair and substitute the x and y values into the inequality. If the resulting statement is true, then the ordered pair is a solution. If the statement is false, then it's not a solution. This method is straightforward and reliable, making it a great tool for tackling these types of problems. Remember, understanding the basics is key to mastering more complex concepts in mathematics. By grasping the meaning of ordered pairs and inequalities, you'll be well-equipped to solve a wide range of problems.
Testing Ordered Pair A: (-3, 0)
Let's start with the first ordered pair, A. (-3, 0). To determine if this pair is a solution to the inequality 2x + y > -4, we need to substitute the x and y values into the inequality and see if the resulting statement is true. In this case, x = -3 and y = 0. So, we'll replace x with -3 and y with 0 in the inequality.
Substituting these values, we get: 2(-3) + 0 > -4. Now, let's simplify the left side of the inequality. 2 multiplied by -3 is -6, so the inequality becomes -6 + 0 > -4. Adding 0 to -6 doesn't change the value, so we have -6 > -4. Now, we need to ask ourselves, is -6 greater than -4? On the number line, -6 is to the left of -4, which means it's a smaller value. Therefore, the statement -6 > -4 is false.
Since the inequality is not true when we substitute the values from the ordered pair (-3, 0), we can conclude that this ordered pair is not a solution to the inequality 2x + y > -4. This process of substitution is crucial in solving inequalities. It allows us to check whether a specific ordered pair satisfies the given condition. We'll repeat this process for each of the remaining ordered pairs to find the solutions. Remember, it's important to pay close attention to the signs and perform the arithmetic accurately to avoid errors. With practice, you'll become more confident and efficient in solving these types of problems.
Testing Ordered Pair B: (0, 1)
Now, let's move on to the second ordered pair, B. (0, 1). Just like before, we'll substitute the x and y values into the inequality 2x + y > -4. In this case, x = 0 and y = 1. Replacing x and y with these values, we get 2(0) + 1 > -4.
Let's simplify the left side of the inequality. 2 multiplied by 0 is 0, so the inequality becomes 0 + 1 > -4. Adding 0 and 1 gives us 1, so we have 1 > -4. Now, we need to determine if this statement is true. Is 1 greater than -4? Yes, it is! On the number line, 1 is to the right of -4, indicating that it's a larger value. Therefore, the statement 1 > -4 is true.
Since the inequality holds true when we substitute the values from the ordered pair (0, 1), we can conclude that this ordered pair is a solution to the inequality 2x + y > -4. This is an important step in the process of finding all the solutions. We've identified one ordered pair that satisfies the inequality, and we'll continue testing the remaining pairs to see if there are any other solutions. Remember, each ordered pair is a potential solution, and the substitution method allows us to verify whether it fits the criteria set by the inequality. This methodical approach ensures that we accurately identify all the ordered pairs that make the inequality true.
Testing Ordered Pair C: (-1, -1)
Let's proceed to the third ordered pair, C. (-1, -1). As we've done before, we'll substitute the x and y values into the inequality 2x + y > -4. Here, x = -1 and y = -1. Substituting these values, we get 2(-1) + (-1) > -4.
Now, let's simplify the left side of the inequality. 2 multiplied by -1 is -2, so the inequality becomes -2 + (-1) > -4. Adding -2 and -1 gives us -3, so we have -3 > -4. We need to determine if this statement is true. Is -3 greater than -4? Yes, it is! On the number line, -3 is to the right of -4, which means it's a larger value. Therefore, the statement -3 > -4 is true.
Since the inequality holds true when we substitute the values from the ordered pair (-1, -1), we can conclude that this ordered pair is also a solution to the inequality 2x + y > -4. We're building up our list of solutions, and it's great to see how the substitution method helps us identify them. Remember, each ordered pair represents a point on the coordinate plane, and we're essentially finding the points that lie in the region defined by the inequality. This visual representation can be helpful in understanding the concept of inequalities. Let's continue testing the remaining ordered pairs to complete our solution set.
Testing Ordered Pair D: (5, -12)
Moving on to the fourth ordered pair, D. (5, -12), we'll follow the same procedure. Substitute the x and y values into the inequality 2x + y > -4. In this case, x = 5 and y = -12. Substituting these values, we get 2(5) + (-12) > -4.
Let's simplify the left side of the inequality. 2 multiplied by 5 is 10, so the inequality becomes 10 + (-12) > -4. Adding 10 and -12 gives us -2, so we have -2 > -4. Now, we need to determine if this statement is true. Is -2 greater than -4? Yes, it is! On the number line, -2 is to the right of -4, indicating that it's a larger value. Therefore, the statement -2 > -4 is true.
Since the inequality holds true when we substitute the values from the ordered pair (5, -12), we can conclude that this ordered pair is also a solution to the inequality 2x + y > -4. Our list of solutions is growing, and we're getting closer to identifying all the ordered pairs that satisfy the inequality. This methodical approach of substituting values and simplifying is a powerful technique in algebra. It allows us to break down complex problems into smaller, manageable steps. Let's proceed to the final ordered pair and complete our analysis.
Testing Ordered Pair E: (4, -12)
Finally, let's test the last ordered pair, E. (4, -12). We'll substitute the x and y values into the inequality 2x + y > -4. Here, x = 4 and y = -12. Substituting these values, we get 2(4) + (-12) > -4.
Let's simplify the left side of the inequality. 2 multiplied by 4 is 8, so the inequality becomes 8 + (-12) > -4. Adding 8 and -12 gives us -4, so we have -4 > -4. Now, we need to determine if this statement is true. Is -4 greater than -4? No, it is not! -4 is equal to -4, but it's not greater than itself. Therefore, the statement -4 > -4 is false.
Since the inequality is not true when we substitute the values from the ordered pair (4, -12), we can conclude that this ordered pair is not a solution to the inequality 2x + y > -4. We've now tested all the ordered pairs and identified which ones satisfy the inequality. This comprehensive approach ensures that we have a complete and accurate solution set. Remember, understanding the nuances of inequalities, such as the difference between greater than (>) and greater than or equal to (≥), is crucial for solving these types of problems correctly.
Conclusion: Identifying the Solutions
After testing each ordered pair, we've successfully identified the solutions to the inequality 2x + y > -4. Let's recap our findings. We tested the following ordered pairs:
- A. (-3, 0): Not a solution
- B. (0, 1): Solution
- C. (-1, -1): Solution
- D. (5, -12): Solution
- E. (4, -12): Not a solution
Therefore, the ordered pairs that are solutions to the inequality 2x + y > -4 are (0, 1), (-1, -1), and (5, -12). We arrived at this conclusion by systematically substituting the x and y values from each ordered pair into the inequality and determining whether the resulting statement was true. This method is a fundamental tool in algebra and is applicable to a wide range of inequality problems.
Understanding how to solve inequalities is crucial for many areas of mathematics, including graphing linear inequalities, solving systems of inequalities, and even more advanced topics like linear programming. The ability to accurately identify solutions is a valuable skill that will serve you well in your mathematical journey. Remember to practice these types of problems regularly to reinforce your understanding and build your confidence. By mastering the basics, you'll be well-prepared to tackle more complex challenges in the future.
For further exploration and practice with inequalities, you can visit resources like Khan Academy's Linear Inequalities section.
