Solving Linear Inequalities Identifying Solutions To Y Less Than 0.5x Plus 2

Linear inequalities, a fundamental concept in algebra, play a crucial role in various mathematical and real-world applications. Understanding how to identify solutions to these inequalities is essential for problem-solving and decision-making. In this article, we will delve into the process of determining which points satisfy the linear inequality $y < 0.5x + 2$. We will explore the graphical representation of this inequality, learn how to test points for validity, and ultimately select three options that meet the criteria. Let's embark on this mathematical journey together!

Understanding Linear Inequalities

Before we jump into solving the specific problem, let's take a moment to grasp the core concept of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which have a single solution or a set of solutions that lie on a line, linear inequalities have a range of solutions that can be represented graphically as a shaded region on a coordinate plane. This region encompasses all the points that satisfy the inequality.

The inequality $y < 0.5x + 2$ represents a region below the line $y = 0.5x + 2$. The line itself is not included in the solution set because the inequality is strictly less than. To visualize this, imagine drawing the line $y = 0.5x + 2$ on a graph. The shaded area below this line represents all the points (x, y) where the y-coordinate is less than 0.5 times the x-coordinate plus 2. Understanding this graphical representation is key to identifying solutions to the inequality.

To determine whether a specific point is a solution to the inequality, we substitute the x and y coordinates of the point into the inequality. If the inequality holds true, then the point is a solution; otherwise, it is not. This process is straightforward but requires careful attention to detail, especially when dealing with negative numbers. In the following sections, we will apply this method to the given points to identify the solutions to the inequality $y < 0.5x + 2$.

Graphical Representation of $y < 0.5x + 2$

To fully grasp the concept of solutions to the inequality $y < 0.5x + 2$, visualizing its graphical representation is immensely helpful. The equation $y = 0.5x + 2$ represents a straight line on the coordinate plane. The inequality $y < 0.5x + 2$ corresponds to the region below this line. Let's break down how we can visualize this:

1. The Line: The equation $y = 0.5x + 2$ is in slope-intercept form ($y = mx + b$), where m represents the slope and b represents the y-intercept. In this case, the slope is 0.5 (which can be seen as a rise of 1 unit for every 2 units of run) and the y-intercept is 2 (meaning the line crosses the y-axis at the point (0, 2)).
2. Plotting the Line: To draw the line, we can plot the y-intercept (0, 2) and then use the slope to find another point. For instance, moving 2 units to the right from (0, 2) and 1 unit up gives us the point (2, 3). Connecting these two points will draw the line. Since the inequality is strictly less than, we draw a dashed line to indicate that points on the line itself are not included in the solution set.
3. The Shaded Region: The area below the dashed line represents all the points (x, y) that satisfy the inequality $y < 0.5x + 2$. This is because for any point in this region, the y-coordinate is less than the corresponding y-value on the line for the same x-coordinate. We can shade this area to visually represent the solution set.

This graphical representation provides a clear understanding of the solutions. Any point that falls within the shaded region is a solution to the inequality. In the next section, we will explore how to test specific points to determine if they lie within this solution region.

Testing Points for Solutions

Now that we understand the graphical representation of the inequality $y < 0.5x + 2$, we can move on to the practical step of testing whether specific points are solutions. The process involves substituting the x and y coordinates of a given point into the inequality and checking if the inequality holds true.

Here are the points we need to evaluate:

• (-3, -2)
• (-2, 1)
• (-1, -2)
• (-1, 2)
• (1, -2)

Let's go through each point systematically:

1. Point (-3, -2): Substitute x = -3 and y = -2 into the inequality:

   -2 < 0.5(-3) + 2 -2 < -1.5 + 2 -2 < 0.5

   This statement is true. Therefore, the point (-3, -2) is a solution.

2. Point (-2, 1): Substitute x = -2 and y = 1 into the inequality:

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

   This statement is false because 1 is not less than 1. Therefore, the point (-2, 1) is not a solution.

3. Point (-1, -2): Substitute x = -1 and y = -2 into the inequality:

   -2 < 0.5(-1) + 2 -2 < -0.5 + 2 -2 < 1.5

   This statement is true. Therefore, the point (-1, -2) is a solution.

4. Point (-1, 2): Substitute x = -1 and y = 2 into the inequality:

   2 < 0.5(-1) + 2 2 < -0.5 + 2 2 < 1.5

   This statement is false. Therefore, the point (-1, 2) is not a solution.

5. Point (1, -2): Substitute x = 1 and y = -2 into the inequality:

   -2 < 0.5(1) + 2 -2 < 0.5 + 2 -2 < 2.5

   This statement is true. Therefore, the point (1, -2) is a solution.

By following this method, we can confidently determine whether any given point is a solution to the linear inequality. In the next section, we will summarize our findings and select the three options that satisfy the given inequality.

Identifying the Solutions: Selecting Three Options

Having meticulously tested each point against the inequality $y < 0.5x + 2$, we can now identify the solutions. Recall that a point is a solution if, upon substituting its coordinates into the inequality, the resulting statement is true. Based on our analysis, the following points satisfy the inequality:

• (-3, -2): The inequality holds true: -2 < 0.5
• (-1, -2): The inequality holds true: -2 < 1.5
• (1, -2): The inequality holds true: -2 < 2.5

On the other hand, the points (-2, 1) and (-1, 2) do not satisfy the inequality, as they result in false statements upon substitution.

Therefore, the three options that are solutions to the linear inequality $y < 0.5x + 2$ are (-3, -2), (-1, -2), and (1, -2). These points lie in the shaded region below the line $y = 0.5x + 2$ on the coordinate plane, which visually represents the solution set of the inequality.

Conclusion

In this comprehensive exploration, we have successfully identified three points that satisfy the linear inequality $y < 0.5x + 2$. We began by understanding the concept of linear inequalities and their graphical representation, which is crucial for visualizing the solution set. We then delved into a step-by-step process of testing points by substituting their coordinates into the inequality. By meticulously evaluating each point, we were able to determine which ones satisfied the given condition.

This exercise highlights the importance of a systematic approach to solving mathematical problems. By combining conceptual understanding with careful execution, we can confidently tackle a wide range of inequality-related challenges. The ability to identify solutions to linear inequalities is a valuable skill that extends beyond the classroom and finds applications in various fields, including economics, engineering, and computer science. As you continue your mathematical journey, remember the principles and techniques we have discussed here, and you will be well-equipped to tackle more complex problems in the future.