Is $(6,4)$ A Solution To $y > -1/2 X + 7$?

Dec 16, 2025 by Andrew McMorgan 43 views

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically tackling inequalities and ordered pairs. You know, those little pairs of numbers like $(6,4)$ that look simple but hold the key to understanding graphical relationships. Our main mission today is to figure out if the ordered pair $(6,4)$ is a solution to the inequality $y>- rac{1}{2} x+7$ . This isn't just about plugging in numbers; it's about understanding what it means for a point to be in the solution set of an inequality. We'll break down why a point might be a solution or not, and explore the visual representation that makes these concepts click.

Understanding Inequalities and Ordered Pairs

First off, let's get our heads around what an ordered pair $(6,4)$ actually represents. In coordinate geometry, an ordered pair $(x,y)$ is like a secret code that tells us a specific location on a graph. The first number, $x$ , tells you how far to move horizontally (left or right from the origin), and the second number, $y$ , tells you how far to move vertically (up or down from the origin). So, $(6,4)$ means we go 6 units to the right and 4 units up.

Now, what about the inequality $y>- rac{1}{2} x+7$ ? This isn't an equation that has a single answer; instead, it represents a region on the graph. It's saying that for any point $(x,y)$ to be considered a solution, the $y$ -value must be greater than the value you get when you plug the $x$ -value into the expression $- rac{1}{2}x+7$ . This 'greater than' symbol ( $>$ ) is crucial here. It means the points that lie exactly on the line represented by $y = - rac{1}{2}x+7$ are not solutions themselves, but they form the boundary of our solution region. The inequality also tells us that the boundary line will be a dashed line, not a solid one, because it's strictly greater than, not greater than or equal to.

So, to determine if our specific point, $(6,4)$ , is a solution to $y>- rac{1}{2} x+7$ , we need to substitute the values of $x$ and $y$ from the ordered pair into the inequality and see if the statement holds true. If it does, the point is a solution. If it doesn't, it's not. It's like a truth test for our coordinates!

Plugging in the Values: The Calculation

Alright, let's get down to business and perform the actual check. We have our ordered pair $(6,4)$ , which means $x=6$ and $y=4$ . Our inequality is $y > - rac{1}{2}x + 7$ . The big question is: Does $4 > - rac{1}{2}(6) + 7$ hold true?

First, we need to calculate the right side of the inequality using $x=6$ :

$- rac{1}{2}(6) + 7$

Multiplying $- rac{1}{2}$ by $6$ gives us $-3$ . So, the expression becomes:

$-3 + 7$

Adding $-3$ and $7$ results in $4$ . So, the right side of the inequality evaluates to $4$ .

Now, we substitute this back into our inequality test:

$4 > 4$

Here's the critical part, guys. Is $4$ greater than $4$ ? Nope! $4$ is equal to $4$ . Since the inequality symbol is strictly 'greater than' ( $>$ ) and not 'greater than or equal to' ( $less$ ), the statement $4 > 4$ is false.

Because the inequality statement is false when we plug in the values from the ordered pair $(6,4)$ , we can definitively say that the ordered pair $(6,4)$ is NOT a solution to the inequality $y>- rac{1}{2} x+7$ . This means that the point $(6,4)$ does not lie within the shaded region that represents all the solutions to this inequality.

Visualizing the Solution: Above or Below the Line?

Now, let's talk about what this means visually on a graph. When we graph the inequality $y>- rac{1}{2} x+7$ , we first consider the line $y = - rac{1}{2}x+7$ . This line acts as a boundary. Since our inequality is strictly greater than ( $>$ ), this boundary line itself is not part of the solution set. We represent this by drawing a dashed line. If the inequality were $less$ or $less$ , we'd draw a solid line because the points on the line would be included.

Our inequality $y>- rac{1}{2} x+7$ indicates that we are looking for all the points $(x,y)$ where the $y$ -coordinate is above the line $y = - rac{1}{2}x+7$ . Think of it this way: for a given $x$ -value, the $y$ -value of the solution points must be higher than the $y$ -value on the line at that same $x$ . This is why we shade the region above the dashed line.

So, where does our point $(6,4)$ fit into this picture? We found that when $x=6$ , the value on the line $y = - rac{1}{2}x+7$ is $y=4$ . This means that the point $(6,4)$ lies exactly on the boundary line $y = - rac{1}{2}x+7$ . Since the boundary line itself is not included in the solution set (because of the strict 'greater than' sign), any point lying on this line cannot be a solution.

Therefore, the statement that $(6,4)$ is a solution is incorrect. It's not a solution because it lies on the boundary line, and the inequality specifically asks for points above the line. If the inequality had been $y less - rac{1}{2}x+7$ , then $(6,4)$ would have been a solution because it would have been on the line, and the line would have been included. But with $y>- rac{1}{2}x+7$ , points on the line are excluded.

To summarize the options provided:

A. Yes, because $(6,4)$ is above the line - Incorrect. Our calculation showed it's not a solution, and visually, it's on the line, not above it. B. No, because $(6,4)$ is below the line - Incorrect. Our calculation showed it's not a solution, but visually, it's on the line, not below it. The region below the line represents $y < - rac{1}{2}x+7$ . C. Yes, because $(6,4)$ is on the line - Incorrect. Being on the line doesn't automatically make it a solution for a strict inequality. D. No, because $(6,4)$ is on the line - Correct. Our calculation proved it's not a solution, and the reason is that it lies on the boundary line, which is excluded by the strict 'greater than' inequality.

Conclusion: Why $(6,4)$ Isn't a Solution

To wrap things up, guys, we've meticulously analyzed the ordered pair $(6,4)$ against the inequality $y>- rac{1}{2} x+7$ . Through direct substitution, we found that $4$ is not greater than $4$ , making the statement false. This means $(6,4)$ does not satisfy the inequality. Visually, when we graph this inequality, the line $y = - rac{1}{2}x+7$ forms a dashed boundary, indicating that points on the line are not included in the solution set. Our point $(6,4)$ coincidentally falls exactly on this boundary line. Therefore, because it lies on the excluded boundary, the ordered pair $(6,4)$ is not a solution to the inequality $y>- rac{1}{2} x+7$ .

The correct option is D. No, because $(6,4)$ is on the line. It's crucial to remember that for strict inequalities ( $<$ or $>$ ), the boundary line is not part of the solution. If the inequality had included 'or equal to' ( $less$ or $less$ ), then points on the line would be solutions. This distinction is key to mastering inequalities. Keep practicing, keep exploring, and you'll be graphing like a pro in no time! Don't forget to check back with Plastik Magazine for more math adventures!