Understanding The Linear Inequality Y > (3/4)x - 2 True Statements And Solutions

In the realm of mathematics, linear inequalities play a crucial role in defining regions and solutions within a coordinate plane. These inequalities, which extend the concept of linear equations, involve comparing two expressions using inequality symbols such as >, <, ≥, and ≤. Understanding the properties and solutions of linear inequalities is fundamental in various mathematical applications, including optimization problems, graphical analysis, and real-world modeling. This article delves into the intricacies of a specific linear inequality, y > (3/4)x - 2, and explores the key statements that accurately describe its characteristics and solutions. By examining the slope, graphical representation, shaded region, and solution points, we aim to provide a comprehensive understanding of this inequality and its significance in the broader context of mathematics.

Understanding the Basics of Linear Inequalities

Before diving into the specific linear inequality, let's establish a solid foundation by reviewing the fundamental concepts. A linear inequality is a mathematical statement that compares two linear expressions using one of the inequality symbols. Unlike linear equations, which represent a straight line on a graph, linear inequalities define a region or half-plane bounded by a line. This boundary line is determined by the corresponding linear equation obtained by replacing the inequality symbol with an equal sign. The solutions to a linear inequality are all the points (x, y) that satisfy the inequality, and these points lie within the shaded region on the graph.

The graphical representation of a linear inequality involves several key elements. First, we graph the boundary line, which is a straight line. If the inequality symbol is > or <, the boundary line is dashed or dotted to indicate that the points on the line are not included in the solution set. If the inequality symbol is ≥ or ≤, the boundary line is solid, indicating that the points on the line are included in the solution set. Next, we shade the region that contains the solutions to the inequality. To determine which region to shade, we can test a point (x, y) that is not on the boundary line. If the point satisfies the inequality, we shade the region containing that point; otherwise, we shade the opposite region. This process of graphing and shading helps visualize the solution set of the linear inequality and provides a clear understanding of the points that satisfy the given condition.

Analyzing the Linear Inequality: y > (3/4)x - 2

Now, let's focus on the specific linear inequality y > (3/4)x - 2 and analyze its properties. This inequality is in slope-intercept form, which is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. In our case, the inequality is y > (3/4)x - 2, so we can identify the slope and y-intercept by comparing it to the slope-intercept form. The coefficient of x is 3/4, which means the slope of the line is 3/4. The constant term is -2, which means the y-intercept is -2. The slope and y-intercept are crucial parameters that define the position and orientation of the boundary line on the coordinate plane. The slope indicates the steepness and direction of the line, while the y-intercept indicates the point where the line crosses the y-axis. Understanding these parameters is essential for accurately graphing the linear inequality and interpreting its solutions.

The boundary line for the linear inequality y > (3/4)x - 2 is the line represented by the equation y = (3/4)x - 2. To graph this line, we can use the slope-intercept form. The y-intercept is -2, so we plot the point (0, -2) on the y-axis. The slope is 3/4, which means for every 4 units we move to the right, we move 3 units up. Starting from the y-intercept, we can find another point on the line by moving 4 units to the right and 3 units up. This gives us the point (4, 1). We can draw a line through these two points to represent the boundary line. Since the inequality symbol is >, the boundary line is dashed, indicating that the points on the line are not included in the solution set. The dashed line serves as a visual separator between the region that satisfies the inequality and the region that does not. The choice of using a dashed or solid line is critical in accurately representing the solution set of the linear inequality and ensuring a clear understanding of the points that are included or excluded.

Evaluating the Statements about the Linear Inequality

Now that we have a solid understanding of the linear inequality y > (3/4)x - 2, let's evaluate the given statements to determine which ones are true. The statements are as follows:

A. The slope of the line is -2. B. The graph of y > (3/4)x - 2 is a dashed line. C. The area below the line is shaded. D. One solution to the linear inequality is (0, 0).

Statement A: The slope of the line is -2.

To evaluate this statement, we need to recall the slope-intercept form of a linear equation, which is y = mx + b, where m represents the slope. In the linear inequality y > (3/4)x - 2, the coefficient of x is 3/4, not -2. Therefore, the slope of the line is 3/4, and Statement A is false. Understanding the slope-intercept form is crucial in correctly identifying the slope of a line, and this statement highlights the importance of careful observation and comparison. The slope is a fundamental characteristic of a line, determining its steepness and direction, and misidentifying the slope can lead to incorrect interpretations of the linear inequality.

Statement B: The graph of y > (3/4)x - 2 is a dashed line.

This statement relates to the graphical representation of the linear inequality. As we discussed earlier, the boundary line is dashed if the inequality symbol is > or <, and solid if the inequality symbol is ≥ or ≤. In our case, the inequality symbol is >, which means the boundary line is indeed dashed. This indicates that the points on the line are not included in the solution set. Therefore, Statement B is true. The use of a dashed line is a visual cue that distinguishes between strict inequalities (>, <) and inclusive inequalities (≥, ≤), and it is essential for accurately representing the solution set of a linear inequality.

Statement C: The area below the line is shaded.

To determine whether the area below the line is shaded, we need to consider the inequality symbol and its implications. The inequality y > (3/4)x - 2 means that we are looking for all the points (x, y) where the y-coordinate is greater than (3/4)x - 2. This corresponds to the region above the line, not below. To confirm this, we can test a point above the line, such as (0, 0). Plugging these values into the inequality, we get 0 > (3/4)(0) - 2, which simplifies to 0 > -2. This is a true statement, indicating that the point (0, 0) is a solution to the inequality, and therefore, the area above the line should be shaded. Therefore, Statement C is false. The direction of shading is determined by the inequality symbol, and it is crucial to correctly identify the region that satisfies the given condition.

Statement D: One solution to the linear inequality is (0, 0).

To verify this statement, we need to substitute the values x = 0 and y = 0 into the linear inequality y > (3/4)x - 2 and check if the inequality holds true. Substituting the values, we get 0 > (3/4)(0) - 2, which simplifies to 0 > -2. This is a true statement, which means the point (0, 0) is indeed a solution to the linear inequality. Therefore, Statement D is true. Testing points is a fundamental method for verifying solutions to linear inequalities and understanding the region that satisfies the given condition.

Conclusion

In conclusion, after analyzing the linear inequality y > (3/4)x - 2 and evaluating the given statements, we have determined that Statements B and D are true, while Statements A and C are false. This exploration has highlighted the key aspects of linear inequalities, including the slope, graphical representation, shaded region, and solution points. Understanding these concepts is essential for solving and interpreting linear inequalities in various mathematical and real-world contexts. By mastering the principles of linear inequalities, we can effectively model and analyze a wide range of scenarios, from optimization problems to graphical analysis.

Final Answer

The true statements about the linear inequality y > (3/4)x - 2 are:

• B. The graph of y > (3/4)x - 2 is a dashed line.
• D. One solution to the linear inequality is (0, 0).