Finding The Ordered Pair Solution For A System Of Inequalities

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#h1 Title: Which Ordered Pair Makes Both Inequalities True?

In the realm of mathematics, solving inequalities is a fundamental skill, particularly when dealing with systems of inequalities. This article delves into the process of identifying ordered pairs that satisfy multiple inequalities simultaneously. We will explore the underlying concepts, step-by-step methods, and practical examples to help you master this essential mathematical technique. Let's start by examining the problem at hand: finding the ordered pair that makes both inequalities true.

Understanding Inequalities and Ordered Pairs

Before diving into the solution, it's crucial to grasp the core concepts of inequalities and ordered pairs. Inequalities, unlike equations, express a relationship where one value is not necessarily equal to another. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Ordered pairs, represented as (x, y), denote a specific point on a coordinate plane, where 'x' is the horizontal coordinate (abscissa) and 'y' is the vertical coordinate (ordinate). When we seek an ordered pair that satisfies an inequality, we are essentially looking for a point that, when its coordinates are substituted into the inequality, makes the inequality a true statement. For instance, in the inequality y < 3x - 1, we want to find an (x, y) pair where the y-value is strictly less than 3 times the x-value minus 1. Similarly, for y ≥ -x + 4, we need an ordered pair where the y-value is greater than or equal to the negative of the x-value plus 4. The challenge arises when we need to find an ordered pair that simultaneously satisfies multiple inequalities, forming a system of inequalities. This involves identifying the region on the coordinate plane where the solutions of all inequalities overlap. To tackle such problems effectively, we employ a combination of algebraic manipulation and graphical analysis, ensuring that the chosen ordered pair lies within the feasible region defined by the inequalities.

The Given Inequalities

Let's consider the specific system of inequalities presented:

y < 3x - 1
y ≥ -x + 4

Our objective is to determine which ordered pair from the given options satisfies both of these inequalities. This means that when we substitute the x and y values of the ordered pair into each inequality, both inequalities must hold true. The first inequality, y < 3x - 1, represents a region on the coordinate plane where the y-coordinate is less than 3 times the x-coordinate minus 1. The boundary line for this inequality is y = 3x - 1, which is a line with a slope of 3 and a y-intercept of -1. Since the inequality is 'less than', the solution set lies below this line, and the line itself is not included (represented by a dashed line when graphing). The second inequality, y ≥ -x + 4, represents a region where the y-coordinate is greater than or equal to the negative of the x-coordinate plus 4. The boundary line here is y = -x + 4, a line with a slope of -1 and a y-intercept of 4. Because the inequality includes 'greater than or equal to', the solution set lies above this line, and the line itself is included (represented by a solid line when graphing). The solution to the system of inequalities is the region where these two individual solution sets overlap. To find an ordered pair that satisfies both inequalities, we need to identify a point that lies within this overlapping region. This can be done algebraically by substituting the coordinates of the given options into the inequalities or graphically by plotting the inequalities and visually identifying the overlapping region. In the subsequent sections, we will explore how to test the given ordered pairs to determine which one satisfies both inequalities.

Testing the Ordered Pairs

Now, let's evaluate the provided ordered pairs to determine which one satisfies both inequalities:

A. (4, 0)

Substitute x = 4 and y = 0 into the inequalities:

  • y < 3x - 1 becomes 0 < 3(4) - 1, which simplifies to 0 < 11. This inequality is true.
  • y ≥ -x + 4 becomes 0 ≥ -4 + 4, which simplifies to 0 ≥ 0. This inequality is also true.

Since both inequalities hold true for (4, 0), this ordered pair is a potential solution.

B. (1, 2)

Substitute x = 1 and y = 2 into the inequalities:

  • y < 3x - 1 becomes 2 < 3(1) - 1, which simplifies to 2 < 2. This inequality is false.

Since the first inequality is false, we don't need to check the second one. The ordered pair (1, 2) is not a solution.

C. (0, 4)

Substitute x = 0 and y = 4 into the inequalities:

  • y < 3x - 1 becomes 4 < 3(0) - 1, which simplifies to 4 < -1. This inequality is false.

As the first inequality is false, the ordered pair (0, 4) is not a solution.

From our analysis, only the ordered pair (4, 0) satisfies both inequalities. This methodical approach of substituting the coordinates into the inequalities allows us to accurately determine the solution set. In the next section, we will solidify our understanding by summarizing the solution and highlighting the importance of this method in solving mathematical problems.

Solution

After evaluating each ordered pair, we've determined that only (4, 0) satisfies both inequalities:

y < 3x - 1
y ≥ -x + 4

For the ordered pair (4, 0):

  • The first inequality, y < 3x - 1, becomes 0 < 3(4) - 1, which simplifies to 0 < 11. This statement is true.
  • The second inequality, y ≥ -x + 4, becomes 0 ≥ -4 + 4, which simplifies to 0 ≥ 0. This statement is also true.

Since both inequalities are true when x = 4 and y = 0, the ordered pair (4, 0) is the solution. The other ordered pairs, (1, 2) and (0, 4), did not satisfy both inequalities. This process of substituting the x and y values into the inequalities and verifying the resulting statements is a fundamental technique in solving systems of inequalities. It ensures that we identify the specific points that lie within the solution set, which is the region where all inequalities are simultaneously satisfied. In this case, the point (4, 0) lies within the overlapping region defined by the two inequalities, making it the correct solution. Understanding and applying this method is crucial for tackling a wide range of mathematical problems involving inequalities and systems of inequalities.

Conclusion

In conclusion, determining which ordered pair satisfies a system of inequalities involves a systematic approach of substituting the coordinates of each pair into the inequalities and verifying if the resulting statements are true. For the given system of inequalities:

y < 3x - 1
y ≥ -x + 4

We found that the ordered pair (4, 0) is the only one that satisfies both inequalities. This process highlights the importance of understanding the meaning of inequalities and how they define regions on the coordinate plane. When dealing with a system of inequalities, the solution is the set of all points that lie in the region where the solutions of each individual inequality overlap. This can be visualized graphically by plotting the inequalities and identifying the common region. Alternatively, as demonstrated in this article, we can use an algebraic approach by substituting the coordinates of potential solutions into the inequalities. This method is particularly useful when dealing with multiple choice questions or when a graphical approach is not feasible. Mastering the techniques for solving systems of inequalities is a crucial skill in mathematics, with applications in various fields such as optimization, linear programming, and economics. By understanding the underlying concepts and practicing the methods, you can confidently tackle a wide range of problems involving inequalities and ordered pairs. Remember, the key is to carefully substitute the values, simplify the expressions, and verify that all inequalities hold true for the selected ordered pair.