Finding Ordered Pairs That Satisfy Inequalities Y ≤ -x + 1 And Y > X

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In mathematics, particularly in algebra, solving systems of inequalities is a fundamental skill. This article delves into a step-by-step approach to determine which ordered pair makes both inequalities true. We will dissect the process, offering clear explanations and illustrative examples to ensure a thorough understanding. Our focus will be on the inequalities y ≤ -x + 1 and y > x , but the methods discussed can be applied to various inequality systems.

Understanding Inequalities and Ordered Pairs

Before we dive into solving the specific system, let's establish a firm grasp of the basic concepts. Inequalities, unlike equations, express a range of possible values rather than a single solution. The symbols used in inequalities are: less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). An ordered pair, represented as (x, y), signifies a point on a coordinate plane. The 'x' value represents the horizontal position, and the 'y' value represents the vertical position. When we say an ordered pair "satisfies" an inequality, it means that substituting the x and y values into the inequality results in a true statement. For instance, if we have the inequality y > x , the ordered pair (2, 3) satisfies it because 3 is indeed greater than 2. However, the ordered pair (1, 0) does not satisfy it because 0 is not greater than 1.

To determine if an ordered pair satisfies a system of inequalities, we must check if it satisfies every inequality in the system. If the ordered pair fails to satisfy even one inequality, it is not a solution to the system. This leads us to the process of graphically representing inequalities and identifying the region of solutions.

Graphical Representation of Inequalities

A graphical representation is a powerful tool for visualizing the solutions to inequalities. Each inequality can be plotted on a coordinate plane, with the solution set represented by a shaded region. The boundary line of this shaded region is determined by the equation formed by replacing the inequality sign with an equality sign. For example, the inequality y ≤ -x + 1 becomes the equation y = -x + 1 , which represents a straight line. This line serves as the boundary. If the inequality includes "≤" or "≥", the boundary line is solid, indicating that points on the line are included in the solution. If the inequality uses "<" or ">", the boundary line is dashed, meaning points on the line are not part of the solution.

To determine which side of the line should be shaded, we choose a test point that is not on the line itself. A common choice is the origin (0, 0), unless the line passes through the origin. We substitute the coordinates of the test point into the original inequality. If the inequality holds true, we shade the side of the line containing the test point. If the inequality is false, we shade the opposite side. This shaded region represents all the ordered pairs that satisfy the inequality. When dealing with a system of inequalities, we graph each inequality separately. The region where the shaded areas of all inequalities overlap represents the solution set for the entire system. Any ordered pair within this overlapping region will satisfy all the inequalities in the system.

Solving the System: y ≤ -x + 1 and y > x

Now, let's apply this knowledge to the specific system of inequalities: y ≤ -x + 1 and y > x . First, we'll graph each inequality individually. For y ≤ -x + 1 , we begin by graphing the line y = -x + 1 . This is a linear equation with a slope of -1 and a y-intercept of 1. We plot the y-intercept at (0, 1) and use the slope to find another point. For instance, moving one unit to the right and one unit down gives us the point (1, 0). We draw a solid line through these points because the inequality includes "≤". To determine the shaded region, we use the test point (0, 0). Substituting into the inequality, we get 0 ≤ -0 + 1, which simplifies to 0 ≤ 1. This is true, so we shade the region below the line, including the line itself.

Next, we graph the inequality y > x . The boundary line is y = x , which is a line that passes through the origin with a slope of 1. We plot the origin (0, 0) and use the slope to find another point, such as (1, 1). We draw a dashed line through these points because the inequality includes ">", meaning points on the line are not included in the solution. Using the test point (0, 0) is not possible here because it lies on the line y = x . So, we choose another test point, such as (0, 1). Substituting into the inequality, we get 1 > 0, which is true. Therefore, we shade the region above the dashed line.

The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This region represents all ordered pairs (x, y) that satisfy both y ≤ -x + 1 and y > x .

Identifying the Correct Ordered Pair

Once we have the graphical representation, we can visually inspect ordered pairs to see if they fall within the overlapping region. However, a more precise method is to substitute the x and y values of each ordered pair into both inequalities and check if they hold true. Let's consider a few examples to illustrate this process.

Suppose we are given the ordered pairs (0, 0), (-1, 0), (0, 1), and (-1, 1). We will test each pair:

  1. (0, 0):

    • For y ≤ -x + 1 : 0 ≤ -0 + 1, which simplifies to 0 ≤ 1. This is true.
    • For y > x : 0 > 0. This is false. Since (0, 0) does not satisfy both inequalities, it is not a solution.

  2. (-1, 0):

    • For y ≤ -x + 1 : 0 ≤ -(-1) + 1, which simplifies to 0 ≤ 2. This is true.
    • For y > x : 0 > -1. This is true. The ordered pair (-1, 0) satisfies both inequalities, making it a solution.

  3. (0, 1):

    • For y ≤ -x + 1 : 1 ≤ -0 + 1, which simplifies to 1 ≤ 1. This is true.
    • For y > x : 1 > 0. This is true. The ordered pair (0, 1) also satisfies both inequalities, making it a solution.

  4. (-1, 1):

    • For y ≤ -x + 1 : 1 ≤ -(-1) + 1, which simplifies to 1 ≤ 2. This is true.
    • For y > x : 1 > -1. This is true. The ordered pair (-1, 1) is yet another solution as it satisfies both inequalities.

By systematically substituting the coordinates of each ordered pair into the inequalities, we can accurately determine which pairs are solutions to the system. This method is particularly useful when dealing with multiple ordered pairs or when a graphical representation is not readily available.

Common Mistakes and How to Avoid Them

Solving systems of inequalities can sometimes be tricky, and it's easy to make common mistakes. One frequent error is shading the wrong region after graphing the boundary line. Always double-check by using a test point and verifying if the inequality holds true. Another mistake is forgetting that a dashed line indicates that points on the line are not included in the solution, while a solid line means they are included. When substituting ordered pairs, ensure you are substituting the x and y values correctly into each inequality. A simple mix-up can lead to an incorrect conclusion.

Furthermore, be meticulous with arithmetic operations. A small calculation error can change the outcome significantly. It's also crucial to remember that an ordered pair must satisfy all inequalities in the system to be considered a solution. Failing to check even one inequality can result in an incorrect answer. By being aware of these potential pitfalls and taking the time to carefully review each step, you can minimize errors and confidently solve systems of inequalities.

Conclusion

Determining which ordered pair makes both inequalities true involves a combination of graphical and algebraic techniques. By understanding the concepts of inequalities, ordered pairs, and graphical representations, we can systematically solve these problems. Graphing the inequalities provides a visual understanding of the solution set, while substituting ordered pairs into the inequalities offers a precise method for verification. By practicing these steps and being mindful of common mistakes, you can master the skill of solving systems of inequalities and confidently tackle related mathematical challenges. The process of identifying solutions to systems of inequalities is a cornerstone of algebra and has widespread applications in various fields, making it a valuable skill to develop and refine. Remember, the key to success lies in a thorough understanding of the fundamentals and consistent practice. With each problem solved, your confidence and proficiency will grow, paving the way for more advanced mathematical explorations.