Solving Inequalities: Find Coordinate Pair Solutions

Hey guys! Today, we're diving into the exciting world of inequalities and coordinate pairs. We've got a problem on our hands: figuring out which coordinate pairs are solutions to the inequality $2x + y < 8$. Don't worry; it's not as intimidating as it sounds! 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 Coordinate Pairs

Before we jump into solving the inequality, let's make sure we're all on the same page with the basics. An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). In our case, we have $2x + y < 8$, which means we're looking for values of $x$ and $y$ that make the expression $2x + y$ less than 8.

A coordinate pair, on the other hand, is simply a set of two numbers, written as $(x, y)$, that represent a point on a coordinate plane. The first number, $x$, tells us how far to move horizontally from the origin (the point where the axes meet), and the second number, $y$, tells us how far to move vertically. For example, the coordinate pair $(3, 2)$ means we move 3 units to the right and 2 units up from the origin.

Now, the big question is: how do we figure out which coordinate pairs satisfy our inequality? The answer is simple: we plug them in! We substitute the $x$ and $y$ values from each coordinate pair into the inequality and see if the statement is true. If it is, then that coordinate pair is a solution. If not, then it's not a solution. Think of it like a puzzle – we're trying to find the pieces that fit!

Let's look at why this process works so well. When we substitute the values of $x$ and $y$ into the inequality, we are essentially testing whether the point represented by the coordinate pair lies in the region of the coordinate plane that satisfies the inequality. The inequality $2x + y < 8$ defines a region on the plane, and any point within that region (or on the boundary if the inequality includes ≤ or ≥) is a solution. By plugging in the coordinates, we're checking if the point falls within this solution region. This method is powerful because it provides a direct way to verify solutions, turning an abstract inequality into a concrete check.

Furthermore, understanding the graphical representation of inequalities can provide an intuitive grasp of the solutions. The inequality $2x + y < 8$ can be visualized as a half-plane on the coordinate grid. The boundary line is determined by the equation $2x + y = 8$, and the inequality represents all the points on one side of this line (but not the line itself, since we have a strict inequality). Testing coordinate pairs is equivalent to plotting these points on the plane and visually checking if they fall within the correct region. This visual approach helps solidify the concept and makes it easier to remember why certain points are solutions and others are not. For instance, if we were to graph the line $2x + y = 8$, we would see that the region where $2x + y < 8$ is the area below the line. Points in this area, when their coordinates are plugged into the inequality, will make the statement true.

Step-by-Step Solution

Okay, let's roll up our sleeves and solve this thing! We have the inequality $2x + y < 8$, and we need to test the following coordinate pairs:

• A. $(0,0)$
• B. $(5,-2)$
• C. $(-1,4)$
• D. $(-3,-5)$
• E. $(7,7)$

We'll go through each option one by one, plugging in the $x$ and $y$ values and checking if the inequality holds true.

Option A: $(0,0)$

Let's substitute $x = 0$ and $y = 0$ into the inequality:

$2(0) + 0 < 8$

$0 + 0 < 8$

$0 < 8$

This statement is true! So, $(0,0)$ is a solution to the inequality.

Option B: $(5,-2)$

Now, let's try $x = 5$ and $y = -2$:

$2(5) + (-2) < 8$

$10 - 2 < 8$

$8 < 8$

This statement is false! 8 is not less than 8. So, $(5,-2)$ is not a solution.

Option C: $(-1,4)$

Next up, $x = -1$ and $y = 4$:

$2(-1) + 4 < 8$

$-2 + 4 < 8$

$2 < 8$

This statement is true! $(-1,4)$ is a solution.

Option D: $(-3,-5)$

Let's plug in $x = -3$ and $y = -5$:

$2(-3) + (-5) < 8$

$-6 - 5 < 8$

$-11 < 8$

This statement is true! $(-3,-5)$ is a solution.

Option E: $(7,7)$

Finally, let's try $x = 7$ and $y = 7$:

$2(7) + 7 < 8$

$14 + 7 < 8$

$21 < 8$

This statement is false! 21 is definitely not less than 8. So, $(7,7)$ is not a solution.

Final Answer and Key Takeaways

Alright, guys, we've done it! We've tested all the coordinate pairs, and here are the solutions to the inequality $2x + y < 8$:

• A. $(0,0)$
• C. $(-1,4)$
• D. $(-3,-5)$

So, the correct answers are A, C, and D. Pat yourselves on the back – you've just tackled an inequality problem like pros!

The main takeaway here is that to check if a coordinate pair is a solution to an inequality, you simply substitute the $x$ and $y$ values into the inequality and see if the resulting statement is true. This method works for any inequality, no matter how complicated it may seem.

Another important point to remember is that inequalities represent a range of solutions, not just a single value like equations often do. In this case, we found three coordinate pairs that work, but there are infinitely many more! This is because the inequality $2x + y < 8$ defines a region on the coordinate plane, and any point within that region is a solution. This is a fundamental concept in mathematics, and it's crucial for understanding more advanced topics like linear programming and systems of inequalities.

Thinking about it graphically can also give you a better understanding. If you were to plot these points on a graph, you’d notice that (0,0), (-1,4), and (-3,-5) would all fall on the same side of the line $2x + y = 8$, which represents the boundary for the inequality. The point (5,-2) and (7,7), which are not solutions, would be on the other side or on the line itself. Visualizing this helps to make the concept more concrete and easier to remember. This connection between algebra and geometry is something that mathematicians use all the time, so it’s a good habit to develop!

Wrapping Up

I hope this breakdown has been helpful and that you now feel confident in your ability to solve inequalities with coordinate pairs. Remember, the key is to take it step by step, plug in the values, and see if the statement holds true. Keep practicing, and you'll become a master of inequalities in no time! Keep an eye out for more math adventures, and don't hesitate to ask questions if anything is unclear. You guys are awesome, and I'm cheering you on every step of the way!