Solutions To The Linear Inequality Y < 0.5x + 2 Explained

Hey guys! Today, we're diving into the world of linear inequalities and figuring out which points are the true MVPs, the ones that actually satisfy the condition $y < 0.5x + 2$. It's like a fun puzzle where we get to test out different coordinates and see if they fit the mold. So, let's put on our detective hats and get started!

Understanding Linear Inequalities

Before we jump into solving, let's break down what a linear inequality actually is. Think of it as a cousin of the linear equation, but instead of an equals sign (=), we've got inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). In our case, we're dealing with the < symbol, meaning we're looking for points where the y-coordinate is strictly less than what we get when we plug the x-coordinate into the expression 0.5x + 2.

Imagine a straight line drawn on a graph. That line represents the equation $y = 0.5x + 2$. Now, the inequality $y < 0.5x + 2$ represents all the points on one side of that line – specifically, the side where the y-values are smaller. It's like we're defining a whole region of the graph, not just a single line. This region is what we call the solution set of the inequality.

To find out if a point is a solution, we simply substitute its x and y coordinates into the inequality and see if the statement holds true. If it does, then the point is part of the solution set; if not, it's out! We have five points to investigate today: (-3, -2), (-2, 1), (-1, -2), (-1, 2), and (1, -2). By testing each point, we can find the three options that are solutions to the given linear inequality. Let's get started with the first point, (-3,-2).

Point-by-Point Verification

Let’s get our hands dirty and test each point one by one. This is where the fun begins, guys! We're going to plug in the x and y values of each point into our inequality $y < 0.5x + 2$ and see if the statement holds water. If it does, bingo, we've found a solution! If not, we move on to the next suspect.

Point 1: (-3, -2)

First up, we've got the point (-3, -2). This means x = -3 and y = -2. Let’s substitute these values into our inequality:

$-2 < 0.5(-3) + 2$

Now, let's simplify the right side of the inequality:

$-2 < -1.5 + 2$

$-2 < 0.5$

Is this statement true? Absolutely! -2 is indeed less than 0.5. So, the point (-3, -2) is a solution to our inequality. That's one down, two more to find! Remember, guys, we are looking for three options, so let's continue our verification.

Point 2: (-2, 1)

Next in line is the point (-2, 1). Here, x = -2 and y = 1. Let’s plug these values into the inequality and see what happens:

$1 < 0.5(-2) + 2$

Simplify the right side:

$1 < -1 + 2$

$1 < 1$

Hold on a second... Is 1 less than 1? Nope! 1 is equal to 1, but it's not less than 1. So, this statement is false, and the point (-2, 1) is not a solution to our inequality. Better luck next time!

Point 3: (-1, -2)

Moving on to our third point, (-1, -2). This time, x = -1 and y = -2. Let's substitute those values in:

$-2 < 0.5(-1) + 2$

Simplify the right side:

$-2 < -0.5 + 2$

$-2 < 1.5$

Is -2 less than 1.5? You bet it is! So, the point (-1, -2) is a solution. We're on a roll, guys! That’s two solutions found, just one more to go.

Point 4: (-1, 2)

Our fourth contender is the point (-1, 2). With x = -1 and y = 2, let's see if it fits the bill:

$2 < 0.5(-1) + 2$

Simplify the right side:

$2 < -0.5 + 2$

$2 < 1.5$

Is 2 less than 1.5? Definitely not! 2 is greater than 1.5, so this statement is false. The point (-1, 2) is not a solution. Keep trying!

Point 5: (1, -2)

Last but not least, we have the point (1, -2). Here, x = 1 and y = -2. Let’s plug them in for the final verdict:

$-2 < 0.5(1) + 2$

Simplify the right side:

$-2 < 0.5 + 2$

$-2 < 2.5$

Is -2 less than 2.5? Absolutely! So, the point (1, -2) is a solution. We’ve found our third solution, guys!

The Triumphant Trio

After carefully testing each point, we've discovered the three points that are solutions to the linear inequality $y < 0.5x + 2$. They are:

• (-3, -2)
• (-1, -2)
• (1, -2)

These points are the VIPs that make our inequality happy. They lie in the region of the graph that satisfies the condition $y < 0.5x + 2$. It was like a fun treasure hunt, wasn't it? We dug through the possibilities and unearthed the solutions, point by point.

Visualizing the Solution

To really drive the point home, imagine graphing the line $y = 0.5x + 2$. It's a straight line with a slope of 0.5 and a y-intercept of 2. Now, the inequality $y < 0.5x + 2$ represents all the points below that line. If you were to plot our three solution points on the graph, you'd see that they all fall neatly into this region below the line.

The points that are not solutions, (-2, 1) and (-1, 2), would either fall on the line itself or above it. This gives us a visual understanding of what it means for a point to satisfy a linear inequality. It's not just about numbers; it's about the relationship between points and lines on a graph.

Why This Matters

Understanding how to solve linear inequalities isn't just a math exercise; it's a powerful tool with real-world applications. Think about scenarios where you have constraints or limitations. For example, you might have a budget and want to know what combinations of items you can afford. Or you might be planning a road trip and need to figure out how far you can drive on a certain amount of gas. Linear inequalities can help you model these situations and find solutions that fit your criteria.

In fields like economics, engineering, and computer science, linear inequalities are used to optimize processes, allocate resources, and make informed decisions. So, mastering this concept opens doors to a whole world of problem-solving possibilities.

Final Thoughts

So there you have it, guys! We've successfully navigated the world of linear inequalities and identified the points that are true solutions to the inequality $y < 0.5x + 2$. Remember, the key is to substitute the x and y values, simplify, and see if the resulting statement is true. It's like a mathematical detective game, and you're the star sleuth!

Keep practicing, keep exploring, and you'll become a linear inequality pro in no time. Math can be fun, and the more you understand it, the more you'll appreciate its power and versatility. Now, go out there and conquer those inequalities!