Solution To The Linear Inequality Y Less Than -1/2x + 2

Understanding Linear Inequalities is crucial for solving mathematical problems and real-world applications. Before we dive into the specific question of identifying a solution to the linear inequality y < -1/2x + 2, let's first establish a solid foundation of what linear inequalities are and how they function. Linear inequalities, at their core, are mathematical expressions that compare two values using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike linear equations, which seek to find exact values that make both sides of the equation equal, linear inequalities deal with a range of values that satisfy the given condition. This distinction is essential because it broadens the scope of possible solutions, making it applicable to scenarios where multiple outcomes are permissible. In the context of a coordinate plane, a linear inequality represents a region rather than a specific line. This region is bounded by the line defined by the corresponding linear equation (e.g., y = -1/2x + 2 in our case), but the inequality sign dictates which side of the line constitutes the solution set. For inequalities involving '<' or '>', the boundary line is dashed or dotted, indicating that the points on the line itself are not included in the solution. Conversely, inequalities with '≤' or '≥' use a solid line to show that the points on the line are part of the solution. Understanding these nuances is fundamental for accurately interpreting and solving linear inequalities, as it allows us to visualize the solution set and determine whether a given point satisfies the inequality. This concept is not just limited to academic exercises; it has practical applications in various fields, from economics (e.g., budget constraints) to engineering (e.g., tolerance limits), where understanding the range of possible solutions is critical for decision-making and problem-solving.

Testing points is a fundamental method to determine if they satisfy a linear inequality. In the given problem, we have the inequality y < -1/2x + 2, and we need to check which of the provided points, namely (2,3), (2,1), (3,-2), and (-1,3), are solutions. The process involves substituting the x and y coordinates of each point into the inequality and evaluating whether the resulting statement is true. For a point to be considered a solution, it must make the inequality hold. Let's illustrate this with the first point, (2,3). We substitute x = 2 and y = 3 into the inequality: 3 < -1/2(2) + 2. Simplifying the right side of the inequality, we get 3 < -1 + 2, which further simplifies to 3 < 1. This statement is clearly false, meaning that the point (2,3) is not a solution to the inequality. This process is repeated for each point. For example, when we test the point (2,1), the inequality becomes 1 < -1/2(2) + 2, which simplifies to 1 < 1. This statement is also false, indicating that (2,1) is not a solution. On the other hand, when we substitute the point (3,-2), we get -2 < -1/2(3) + 2, which simplifies to -2 < -1.5 + 2, and further to -2 < 0.5. This statement is true, confirming that the point (3,-2) is indeed a solution to the inequality. Finally, testing the point (-1,3) yields 3 < -1/2(-1) + 2, which simplifies to 3 < 0.5 + 2, and then to 3 < 2.5. This statement is false, meaning that (-1,3) is not a solution. By systematically testing each point, we can confidently identify which ones satisfy the inequality and which ones do not. This method not only provides a definitive answer but also reinforces the understanding of how points relate to inequalities in a coordinate plane. It's a practical approach that can be applied to various inequality problems and is an essential skill in algebra and beyond.

Solving the problem requires a methodical, step-by-step approach. Our main goal is to determine which of the given points satisfies the inequality y < -1/2x + 2. To achieve this, we will systematically substitute the coordinates of each point into the inequality and assess whether the resulting statement is true or false. This process will allow us to identify the point or points that lie within the solution region of the inequality. Let's begin with the first point, (2,3). To test this point, we replace x with 2 and y with 3 in the inequality, yielding 3 < -1/2(2) + 2. Now, we simplify the expression on the right-hand side. Multiplying -1/2 by 2 gives us -1, so the inequality becomes 3 < -1 + 2. Further simplification gives us 3 < 1. This statement is false, as 3 is not less than 1. Therefore, the point (2,3) is not a solution to the inequality. Next, we move on to the second point, (2,1). Substituting x = 2 and y = 1 into the inequality, we get 1 < -1/2(2) + 2. Simplifying the right-hand side again, we have 1 < -1 + 2, which further simplifies to 1 < 1. This statement is also false because 1 is not less than 1. Thus, the point (2,1) is not a solution either. Now, let's test the third point, (3,-2). Substituting x = 3 and y = -2, the inequality becomes -2 < -1/2(3) + 2. Simplifying the right side, we get -2 < -1.5 + 2, which simplifies further to -2 < 0.5. This statement is true, as -2 is indeed less than 0.5. Therefore, the point (3,-2) is a solution to the inequality. Finally, we consider the last point, (-1,3). Substituting x = -1 and y = 3, the inequality becomes 3 < -1/2(-1) + 2. Simplifying the right-hand side, we have 3 < 0.5 + 2, which further simplifies to 3 < 2.5. This statement is false because 3 is not less than 2.5. Therefore, the point (-1,3) is not a solution. Through this methodical, step-by-step process, we have determined that only the point (3,-2) satisfies the given inequality. This comprehensive approach ensures accuracy and clarity in problem-solving, which is crucial for mathematical success.

In conclusion, the point (3,-2) is the only solution among the given options that satisfies the linear inequality y < -1/2x + 2. This determination was made by systematically substituting the coordinates of each point into the inequality and evaluating the resulting statement. This method is a fundamental technique in algebra for verifying solutions to inequalities and understanding their graphical representation. Linear inequalities are a critical concept in mathematics with broad applications in various fields, including economics, engineering, and computer science. They help model situations where relationships are not necessarily equal but fall within a certain range or constraint. For instance, in economics, linear inequalities can represent budget constraints, defining the feasible set of consumption bundles given a limited income. In engineering, they can describe tolerance limits for manufacturing processes, ensuring that products meet certain quality standards. Understanding linear inequalities also lays the groundwork for more advanced mathematical concepts such as linear programming, which is used to optimize solutions in complex systems. Furthermore, the ability to solve and interpret inequalities is essential for data analysis and decision-making in many real-world scenarios. By mastering the techniques discussed, such as substituting points and evaluating inequalities, students and professionals can confidently tackle problems involving constraints and ranges of values. The process of testing points against an inequality not only provides a specific answer but also enhances the understanding of how inequalities define regions in a coordinate plane. This visual and algebraic connection is crucial for developing a strong foundation in mathematical problem-solving. Therefore, a solid grasp of linear inequalities is not just an academic exercise but a valuable skill that empowers individuals to analyze and solve a wide range of practical problems.