Solving Inequalities Identifying Ordered Pair Solutions
Hey everyone! Today, we're diving into a fun mathematical problem where we need to figure out which ordered pair is a solution to a given inequality. Specifically, we'll be tackling the inequality y > (1/4)x + 5. Don't worry, it's not as scary as it looks! We'll break it down step by step, making it super easy to understand. So, grab your thinking caps, and let's get started!
Understanding Inequalities and Ordered Pairs
Before we jump into the solution, let's quickly recap what inequalities and ordered pairs are. An inequality, unlike an equation, doesn't have a single solution but rather a range of solutions. The > symbol means "greater than," so our inequality is saying that y must be greater than the expression (1/4)x + 5. 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, and the second number, y, tells us how far to move vertically.
To find the solution, we have to dive deep into the concept of inequalities and ordered pairs. An inequality is like an equation, but instead of an equals sign (=), it uses symbols like >, <, ≥, or ≤. These symbols show a range of possible values, not just one specific answer. In our case, the inequality y > (1/4)x + 5 means we're looking for all the points where the y-value is bigger than the result of the expression (1/4)x + 5. Imagine a line on a graph – an inequality represents all the points above or below that line, depending on the symbol.
Ordered pairs, like (12, 8) or (4, 7), are the coordinates of a specific point on a graph. The first number (x) tells you how far to go left or right from the center (the origin), and the second number (y) tells you how far to go up or down. When we're solving inequalities, we want to know if a particular ordered pair makes the inequality true. Does the point sit in the shaded area of the graph that represents the solution? That's what we're figuring out!
The Key: Substitution
The key to solving this problem is substitution. We'll take each ordered pair and plug the x and y values into the inequality. If the inequality holds true, then that ordered pair is a solution. If not, it's not a solution. Simple as that!
H2: Testing the Ordered Pairs
Now, let's get our hands dirty and test each of the given ordered pairs:
H3: Ordered Pair (12, 8)
Let's substitute x = 12 and y = 8 into the inequality y > (1/4)x + 5:
8 > (1/4)(12) + 5
First, we need to calculate (1/4)(12). This is the same as 12 divided by 4, which equals 3. So, our inequality now looks like this:
8 > 3 + 5
Next, we add 3 and 5, which gives us 8:
8 > 8
Now, here's the crucial part: Is 8 greater than 8? No, it's not. 8 is equal to 8, but it's not greater. Therefore, the ordered pair (12, 8) is not a solution to the inequality.
H3: Ordered Pair (11, 7)
Next up, let's try the ordered pair (11, 7). We'll substitute x = 11 and y = 7 into the inequality:
7 > (1/4)(11) + 5
Calculating (1/4)(11) gives us 2.75 (since 11 divided by 4 is 2.75). Our inequality now looks like this:
7 > 2.75 + 5
Adding 2.75 and 5, we get 7.75:
7 > 7.75
Is 7 greater than 7.75? Nope! 7 is smaller than 7.75. So, the ordered pair (11, 7) is not a solution either.
H3: Ordered Pair (8, 6)
Let's move on to the ordered pair (8, 6). Substitute x = 8 and y = 6 into the inequality:
6 > (1/4)(8) + 5
Calculating (1/4)(8) is the same as 8 divided by 4, which equals 2. So, the inequality becomes:
6 > 2 + 5
Adding 2 and 5, we get 7:
6 > 7
Is 6 greater than 7? Definitely not! 6 is less than 7. Therefore, the ordered pair (8, 6) is not a solution to our inequality.
H3: Ordered Pair (4, 7)
Finally, let's test the ordered pair (4, 7). We substitute x = 4 and y = 7 into the inequality:
7 > (1/4)(4) + 5
Calculating (1/4)(4) gives us 1 (since 4 divided by 4 is 1). The inequality now looks like this:
7 > 1 + 5
Adding 1 and 5, we get 6:
7 > 6
Is 7 greater than 6? Yes, it is! 7 is indeed larger than 6. Therefore, the ordered pair (4, 7) is a solution to the inequality y > (1/4)x + 5.
H2: The Solution and What We Learned
So, after testing all the ordered pairs, we found that only (4, 7) is a solution to the inequality y > (1/4)x + 5. Hooray! We did it!
In this problem, we learned a super valuable skill: how to determine if an ordered pair is a solution to an inequality. The process is straightforward – substitute the x and y values into the inequality and see if it holds true. This skill is fundamental in algebra and will come in handy in many future math problems.
Key Takeaways:
- Inequalities represent a range of solutions.
- Ordered pairs represent points on a coordinate plane.
- Substitution is the key to checking if an ordered pair is a solution to an inequality.
- Pay close attention to the inequality symbol (>, <, ≥, ≤) to determine if the inequality holds true.
Keep practicing, guys, and you'll become inequality-solving pros in no time! Remember, math is like a puzzle, and each problem is a chance to learn something new.
