Ordered Pair Solution For Inequalities Y>-2x+3 And Y≤x-2
As students delve deeper into the realm of mathematics, the concept of inequalities and their graphical representation becomes increasingly crucial. Understanding how to identify ordered pairs that satisfy multiple inequalities is a fundamental skill in algebra and pre-calculus. This article will explore the intricacies of solving systems of inequalities, focusing on how to determine which ordered pairs make a set of inequalities true. We'll dissect the given problem, providing a step-by-step solution and offering insights into the underlying mathematical principles. Whether you're a student grappling with this topic or an educator seeking to explain it more effectively, this guide will provide clarity and confidence in tackling inequality problems.
Understanding Inequalities and Ordered Pairs
Before diving into the specific problem, let's establish a solid foundation by reviewing the basics of inequalities and ordered pairs. Inequalities, unlike equations, represent a range of values rather than a single solution. They use symbols such as >, <, ≥, and ≤ to indicate whether a value is greater than, less than, greater than or equal to, or less than or equal to another value. When dealing with two-variable inequalities, we're essentially defining regions on a coordinate plane.
An ordered pair, represented as (x, y), is a point on this coordinate plane. To determine if an ordered pair satisfies an inequality, we substitute the x and y values into the inequality and check if the resulting statement is true. For instance, if we have the inequality y > x + 1 and the ordered pair (2, 4), we substitute x = 2 and y = 4 into the inequality, yielding 4 > 2 + 1, which simplifies to 4 > 3. Since this statement is true, the ordered pair (2, 4) satisfies the inequality.
When dealing with a system of inequalities, we need to find ordered pairs that satisfy all the inequalities simultaneously. This means that the ordered pair must lie in the region where the solutions of all inequalities overlap. Graphically, this region is the intersection of the shaded areas representing each inequality. To find such ordered pairs algebraically, we test each option by substituting the x and y values into all the inequalities and checking for consistency.
Understanding these fundamental concepts is crucial for solving the problem at hand and for tackling more complex problems involving inequalities. The ability to interpret inequalities and ordered pairs is not just a mathematical skill but also a valuable tool for problem-solving in various real-world scenarios. For example, in economics, inequalities can represent budget constraints, while in engineering, they can define the safe operating range of a system. Therefore, mastering this topic is an investment in your overall analytical and problem-solving abilities.
Dissecting the Problem: A Step-by-Step Approach
The given problem presents us with two inequalities:
- y > -2x + 3
- y ≤ x - 2
Our task is to identify which of the provided ordered pairs – (0, 0), (0, -1), (1, 1), and (3, 0) – satisfies both inequalities simultaneously. To accomplish this, we will systematically test each ordered pair by substituting its x and y values into both inequalities. This process will reveal whether the ordered pair makes both inequalities true.
Let's start with the first ordered pair, (0, 0). We substitute x = 0 and y = 0 into the first inequality, y > -2x + 3, which gives us 0 > -2(0) + 3. Simplifying this, we get 0 > 3, which is a false statement. Since the first inequality is not satisfied, we can conclude that (0, 0) is not a solution to the system of inequalities. There's no need to test the second inequality for this ordered pair, as it must satisfy both to be a valid solution.
Next, we move on to the second ordered pair, (0, -1). Substituting x = 0 and y = -1 into the first inequality, we get -1 > -2(0) + 3. Simplifying, we have -1 > 3, which is also a false statement. Therefore, (0, -1) does not satisfy the system of inequalities either.
Now, let's consider the ordered pair (1, 1). Substituting x = 1 and y = 1 into the first inequality, we get 1 > -2(1) + 3. Simplifying, we have 1 > 1, which is false. Although this is close to being true, the inequality requires y to be strictly greater than -2x + 3, not equal to it. Since the first inequality is not satisfied, (1, 1) is not a solution.
Finally, we test the ordered pair (3, 0). Substituting x = 3 and y = 0 into the first inequality, we get 0 > -2(3) + 3. Simplifying, we have 0 > -6 + 3, which further simplifies to 0 > -3. This statement is true. Now, we need to check if (3, 0) satisfies the second inequality, y ≤ x - 2. Substituting x = 3 and y = 0, we get 0 ≤ 3 - 2, which simplifies to 0 ≤ 1. This statement is also true. Since (3, 0) satisfies both inequalities, it is the solution we are looking for.
By systematically testing each ordered pair, we have successfully identified the one that makes both inequalities true. This methodical approach is crucial for solving problems involving systems of inequalities, ensuring accuracy and clarity in the solution process.
Detailed Solution and Explanation
To reiterate, the problem asks us to find the ordered pair that satisfies both of the following inequalities:
- y > -2x + 3
- y ≤ x - 2
The given options are:
A. (0, 0) B. (0, -1) C. (1, 1) D. (3, 0)
We will now meticulously examine each option to determine which one fits the criteria.
Option A: (0, 0)
Let's substitute x = 0 and y = 0 into both inequalities:
- For y > -2x + 3: 0 > -2(0) + 3 simplifies to 0 > 3, which is false.
Since the first inequality is not satisfied, we can immediately rule out option A without checking the second inequality.
Option B: (0, -1)
Substitute x = 0 and y = -1 into both inequalities:
- For y > -2x + 3: -1 > -2(0) + 3 simplifies to -1 > 3, which is false.
Again, the first inequality is not satisfied, so option B is not a solution.
Option C: (1, 1)
Substitute x = 1 and y = 1 into both inequalities:
- For y > -2x + 3: 1 > -2(1) + 3 simplifies to 1 > -2 + 3, which further simplifies to 1 > 1. This statement is false because 1 is not greater than 1.
As before, the first inequality is not satisfied, eliminating option C as a possible solution.
Option D: (3, 0)
Substitute x = 3 and y = 0 into both inequalities:
- For y > -2x + 3: 0 > -2(3) + 3 simplifies to 0 > -6 + 3, which further simplifies to 0 > -3. This statement is true.
- For y ≤ x - 2: 0 ≤ 3 - 2 simplifies to 0 ≤ 1. This statement is also true.
Since (3, 0) satisfies both inequalities, it is the correct answer.
Therefore, the ordered pair that makes both inequalities true is (3, 0).
This detailed step-by-step solution demonstrates the importance of systematically testing each option to arrive at the correct answer. By substituting the values of x and y into each inequality and simplifying, we can clearly determine whether an ordered pair is a solution to the system of inequalities.
Graphical Interpretation of the Solution
Beyond the algebraic method we've employed, a graphical approach provides a visual understanding of the solution. Each inequality represents a region on the coordinate plane, and the solution to the system of inequalities is the intersection of these regions. Let's explore how the graphical representation reinforces our solution.
Consider the first inequality, y > -2x + 3. To graph this, we first treat it as an equation, y = -2x + 3, which represents a line with a slope of -2 and a y-intercept of 3. Since the inequality is strictly greater than, we draw a dashed line to indicate that points on the line are not included in the solution. The region above this line represents all the points where y > -2x + 3. This is because for any x-value, the y-values above the line are greater than the corresponding y-value on the line.
Now, let's consider the second inequality, y ≤ x - 2. We treat this as an equation, y = x - 2, which represents a line with a slope of 1 and a y-intercept of -2. Since the inequality includes
