Determine Ordered Pair Solution For Inequalities Y ≤ -x + 1 And Y > X
In mathematics, solving systems of inequalities is a fundamental concept, particularly in algebra and precalculus. When we are presented with a set of inequalities, our goal is often to find the region in the coordinate plane where all inequalities hold true simultaneously. This region represents the set of all ordered pairs (x, y) that satisfy every inequality in the system. In this article, we will delve into how to determine which ordered pair, if any, satisfies a given system of inequalities. Specifically, we will explore the system:
y ≤ -x + 1
y > x
And determine if the ordered pair (-3, 5) satisfies both inequalities. This exploration will not only provide a solution to this particular problem but also offer a comprehensive understanding of the methodologies used in solving similar problems.
Understanding Inequalities and Ordered Pairs
Before we jump into solving the problem, let's first understand the basic concepts involved. Inequalities, unlike equations, do not have a single solution but rather a range of solutions. They use symbols like ≤ (less than or equal to), < (less than), ≥ (greater than or equal to), and > (greater than) to express relationships between values. An ordered pair (x, y) represents a point on the coordinate plane. The x-coordinate tells us how far to move horizontally from the origin (0, 0), and the y-coordinate tells us how far to move vertically.
When we are given an inequality such as y ≤ -x + 1, we are not looking for a single point but rather a region on the coordinate plane. This region includes all points where the y-coordinate is less than or equal to the expression -x + 1. Similarly, for y > x, we are looking for all points where the y-coordinate is greater than the x-coordinate. The solution to a system of inequalities is the overlap, or intersection, of these regions, representing all points that satisfy every inequality in the system. To determine if an ordered pair satisfies a system of inequalities, we substitute the x and y values into each inequality and check if the resulting statements are true. This method is straightforward and effective, providing a clear indication of whether a given point lies within the solution region.
Verifying the Ordered Pair (-3, 5) for y ≤ -x + 1
Let's start by verifying whether the ordered pair (-3, 5) satisfies the first inequality, which is y ≤ -x + 1. To do this, we will substitute x = -3 and y = 5 into the inequality:
5 ≤ -(-3) + 1
Now, we simplify the right side of the inequality:
5 ≤ 3 + 1
5 ≤ 4
The resulting statement is 5 ≤ 4, which is false. This indicates that the ordered pair (-3, 5) does not satisfy the first inequality. Since the ordered pair must satisfy both inequalities to be a solution to the system, we can already conclude that (-3, 5) is not a solution. However, for the sake of completeness and to illustrate the entire process, we will also check the second inequality.
Checking the Ordered Pair (-3, 5) for y > x
Now, let's examine the second inequality, which is y > x. We will again substitute x = -3 and y = 5 into the inequality:
5 > -3
This statement is true because 5 is indeed greater than -3. However, as we determined earlier, the ordered pair (-3, 5) did not satisfy the first inequality. For an ordered pair to be a solution to the system, it must satisfy both inequalities. Since (-3, 5) fails to satisfy y ≤ -x + 1, it cannot be a solution to the system of inequalities.
Methodologies for Solving Systems of Inequalities
Solving systems of inequalities involves finding the set of all ordered pairs that satisfy all inequalities in the system. There are several methodologies to approach this, including graphical methods and algebraic methods. The graphical method provides a visual representation of the solution, while algebraic methods offer a more precise way to identify specific solutions or regions.
Graphical Method
The graphical method involves graphing each inequality on the coordinate plane. For each inequality, we first treat it as an equation and graph the corresponding line. For example, for the inequality y ≤ -x + 1, we would graph the line y = -x + 1. The line divides the plane into two regions, and we need to determine which region satisfies the inequality. If the inequality includes ≤ or ≥, the line is solid, indicating that points on the line are included in the solution. If the inequality includes < or >, the line is dashed, indicating that points on the line are not included.
To determine which region to shade, we can test a point that is not on the line, such as (0, 0). If the point satisfies the inequality, we shade the region containing the point; otherwise, we shade the other region. The solution to the system of inequalities is the region where the shaded areas of all inequalities overlap. This overlapping region represents all the ordered pairs that satisfy every inequality in the system. Graphing the inequalities provides a clear visual representation of the solution set, making it easier to identify the range of possible solutions.
Algebraic Method
Algebraic methods involve manipulating the inequalities to isolate variables or to combine inequalities. One common technique is substitution, where we solve one inequality for one variable and substitute that expression into another inequality. This can help us reduce the system to a single inequality with one variable, which is easier to solve. Another method is elimination, where we add or subtract inequalities to eliminate a variable.
For example, if we have the system:
x + y ≤ 5
x - y > 1
We can add the two inequalities to eliminate y:
(x + y) + (x - y) ≤ 5 + 1
2x ≤ 6
x ≤ 3
This gives us a bound on x. We can then substitute this back into one of the original inequalities to find a bound on y. Algebraic methods are particularly useful when dealing with more complex systems of inequalities or when a precise solution is required.
Combining Graphical and Algebraic Methods
In many cases, combining both graphical and algebraic methods provides the most effective approach to solving systems of inequalities. The graphical method gives a visual overview of the solution region, while algebraic methods allow for precise calculations and verification. For instance, we can use the graphical method to identify the general region where the solution lies and then use algebraic methods to find specific points or boundaries within that region.
Common Mistakes and How to Avoid Them
When working with inequalities, it's crucial to be careful with the rules of algebra, especially when multiplying or dividing by a negative number. One of the most common mistakes is forgetting to flip the inequality sign when multiplying or dividing by a negative number. For example, if we have the inequality -2x < 4, we need to divide both sides by -2, but we must also flip the inequality sign:
-2x < 4
x > -2
Another common mistake is misinterpreting the inequality symbols. Remember that ≤ means “less than or equal to,” and ≥ means “greater than or equal to.” A solid line on a graph indicates that the points on the line are included in the solution, while a dashed line indicates that they are not. Pay close attention to the inequality symbols to ensure you are correctly interpreting and graphing the solution.
When shading regions on a graph, always test a point to ensure you are shading the correct side of the line. This will help you avoid errors in identifying the solution region. Additionally, double-check your work, especially when substituting values into inequalities, to ensure you have not made any arithmetic errors. By being mindful of these common mistakes, you can improve your accuracy and confidence in solving systems of inequalities.
Conclusion
In this article, we addressed the question of whether the ordered pair (-3, 5) satisfies the system of inequalities:
y ≤ -x + 1
y > x
By substituting the values of x and y into the inequalities, we determined that (-3, 5) does not satisfy the first inequality, y ≤ -x + 1, and therefore is not a solution to the system. We also discussed methodologies for solving systems of inequalities, including graphical and algebraic methods, and highlighted common mistakes to avoid. Understanding how to solve systems of inequalities is a critical skill in mathematics, with applications ranging from linear programming to calculus. Mastering these concepts will not only help in academic settings but also in real-world problem-solving scenarios where constraints and conditions need to be considered simultaneously.
By thoroughly understanding the concepts of inequalities and ordered pairs, and by applying the correct methodologies, you can confidently tackle similar problems and expand your mathematical proficiency. Remember to always double-check your work and pay attention to the details to ensure accurate results. Solving systems of inequalities is a valuable skill that opens doors to more advanced mathematical topics and applications.
