Which Ordered Pairs Are In The Solution Set Of The System Of Linear Inequalities?${ \begin{align} y & \geq -\frac{1}{2} X \ y & \ \textless \ \frac{1}{2} X + 1 \end{align} }$A. { (5, -2), (3, 1), (-4, 2)$} B . \[ B. \[ B . \[ (5, -2),

Mar 10, 2025 by ADMIN 240 views

**Which Ordered Pairs are in the Solution Set of the System of Linear Inequalities?**

Introduction

In mathematics, a system of linear inequalities is a set of linear inequalities that are combined to form a single system. The solution set of a system of linear inequalities is the set of all possible solutions that satisfy all the inequalities in the system. In this article, we will explore the solution set of a system of linear inequalities and determine which ordered pairs are in the solution set.

Understanding the System of Linear Inequalities

The given system of linear inequalities is:

\begin{align*} y & \geq -\frac{1}{2} x \\ y & \ \textless \ \frac{1}{2} x + 1 \end{align*}

The first inequality is $y \geq -\frac{1}{2} x$ , which means that $y$ is greater than or equal to $-\frac{1}{2} x$ . The second inequality is $y < \frac{1}{2} x + 1$ , which means that $y$ is less than $\frac{1}{2} x + 1$ .

Graphing the Inequalities

To visualize the solution set, we can graph the two inequalities on a coordinate plane. The first inequality, $y \geq -\frac{1}{2} x$ , can be graphed as a solid line with a slope of $-\frac{1}{2}$ and a y-intercept of $0$ . The second inequality, $y < \frac{1}{2} x + 1$ , can be graphed as a dashed line with a slope of $\frac{1}{2}$ and a y-intercept of $1$ .

Finding the Solution Set

The solution set of the system of linear inequalities is the region where the two inequalities overlap. To find the solution set, we need to find the intersection of the two regions.

The first inequality, $y \geq -\frac{1}{2} x$ , represents a region below the line $y = -\frac{1}{2} x$ . The second inequality, $y < \frac{1}{2} x + 1$ , represents a region below the line $y = \frac{1}{2} x + 1$ .

The intersection of the two regions is the region where both inequalities are satisfied. This region is below the line $y = -\frac{1}{2} x$ and above the line $y = \frac{1}{2} x + 1$ .

Determining the Ordered Pairs

To determine which ordered pairs are in the solution set, we need to check if each ordered pair satisfies both inequalities.

Let's consider the ordered pairs given in the options:

Option A: $(5, -2), (3, 1), (-4, 2)$
Option B: $(5, -2)$

We need to check if each ordered pair satisfies both inequalities.

Checking Option A

Let's check if the ordered pairs in Option A satisfy both inequalities.

$(5, -2)$ : $-2 \geq -\frac{1}{2} (5)$ is true, but $-2 < \frac{1}{2} (5) + 1$ is false.
$(3, 1)$ : $1 \geq -\frac{1}{2} (3)$ is true, but $1 < \frac{1}{2} (3) + 1$ is false.
$(-4, 2)$ : $2 \geq -\frac{1}{2} (-4)$ is true, and $2 < \frac{1}{2} (-4) + 1$ is true.

Only the ordered pair $(-4, 2)$ satisfies both inequalities.

Checking Option B

Let's check if the ordered pair in Option B satisfies both inequalities.

$(5, -2)$ : $-2 \geq -\frac{1}{2} (5)$ is true, but $-2 < \frac{1}{2} (5) + 1$ is false.

The ordered pair $(5, -2)$ does not satisfy both inequalities.

Conclusion

In conclusion, the only ordered pair that is in the solution set of the system of linear inequalities is $(-4, 2)$ . The other ordered pairs given in the options do not satisfy both inequalities.

Final Answer

The final answer is:

Only the ordered pair $(-4, 2)$ is in the solution set of the system of linear inequalities.
Q&A: System of Linear Inequalities =====================================

Introduction

In the previous article, we explored the solution set of a system of linear inequalities and determined which ordered pairs are in the solution set. In this article, we will answer some frequently asked questions about system of linear inequalities.

Q: What is a system of linear inequalities?

A system of linear inequalities is a set of linear inequalities that are combined to form a single system. The solution set of a system of linear inequalities is the set of all possible solutions that satisfy all the inequalities in the system.

Q: How do I graph a system of linear inequalities?

To graph a system of linear inequalities, you need to graph each inequality separately and then find the intersection of the two regions. The solution set of the system is the region where both inequalities are satisfied.

Q: What is the difference between a solid line and a dashed line in a system of linear inequalities?

In a system of linear inequalities, a solid line represents an inequality that is greater than or equal to, while a dashed line represents an inequality that is less than.

Q: How do I determine the solution set of a system of linear inequalities?

To determine the solution set of a system of linear inequalities, you need to find the intersection of the two regions. The solution set is the region where both inequalities are satisfied.

Q: Can a system of linear inequalities have multiple solution sets?

Yes, a system of linear inequalities can have multiple solution sets. This occurs when the two inequalities intersect at multiple points.

Q: How do I check if an ordered pair is in the solution set of a system of linear inequalities?

To check if an ordered pair is in the solution set of a system of linear inequalities, you need to substitute the x and y values of the ordered pair into each inequality and check if the inequality is satisfied.

Q: What is the importance of system of linear inequalities in real-life applications?

System of linear inequalities has many real-life applications, such as:

Budgeting and financial planning
Resource allocation and management
Optimization problems
Game theory and economics

Q: Can a system of linear inequalities be solved using algebraic methods?

Yes, a system of linear inequalities can be solved using algebraic methods, such as substitution and elimination.

Q: What are some common mistakes to avoid when solving a system of linear inequalities?

Some common mistakes to avoid when solving a system of linear inequalities include:

Not graphing the inequalities correctly
Not finding the intersection of the two regions
Not checking if the ordered pair satisfies both inequalities

Conclusion

In conclusion, system of linear inequalities is an important topic in mathematics that has many real-life applications. By understanding the concepts and techniques involved in solving system of linear inequalities, you can apply them to solve problems in various fields.

Final Answer

The final answer is:

System of linear inequalities is a set of linear inequalities that are combined to form a single system.
The solution set of a system of linear inequalities is the set of all possible solutions that satisfy all the inequalities in the system.
To determine the solution set of a system of linear inequalities, you need to find the intersection of the two regions.
System of linear inequalities has many real-life applications, such as budgeting and financial planning, resource allocation and management, optimization problems, and game theory and economics.