Which Ordered Pair Makes Both Inequalities True?${ \begin{align} y &\leq -x + 1 \ y &\ \textgreater \ X \end{align} }$

Mar 10, 2025 by ADMIN 124 views

Solving Inequalities: Finding the Ordered Pair that Satisfies Both Conditions

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions. When dealing with multiple inequalities, it's essential to find the solution that satisfies both conditions. In this article, we'll explore how to solve a system of inequalities and find the ordered pair that makes both inequalities true.

Understanding the Inequalities

The given inequalities are:

$y \leq -x + 1$
$y > x$

The first inequality, $y \leq -x + 1$ , represents a line with a slope of -1 and a y-intercept of 1. The symbol $\leq$ indicates that the line is included in the solution set. The second inequality, $y > x$ , represents a line with a slope of 1 and a y-intercept of 0. The symbol $>$ indicates that the line is not included in the solution set.

Graphing the Inequalities

To visualize the solution set, we can graph the inequalities on a coordinate plane.

Graph of the First Inequality

The graph of the first inequality, $y \leq -x + 1$ , is a line with a slope of -1 and a y-intercept of 1. The line is included in the solution set, so we shade the region below the line.

x	y
-∞	-∞
0	1
∞	-∞

### Graph of the First Inequality x y -∞ -∞ 0 1 ∞ -∞

Graph of the Second Inequality

The graph of the second inequality, $y > x$ , is a line with a slope of 1 and a y-intercept of 0. The line is not included in the solution set, so we shade the region above the line.

x	y
-∞	-∞
0	0
∞	∞

### Graph of the Second Inequality x y -∞ -∞ 0 0 ∞ ∞

Finding the Intersection

To find the ordered pair that satisfies both inequalities, we need to find the intersection of the two shaded regions. The intersection occurs when the two lines intersect, which is at the point (0, 1).

x	y
0	1

### Intersection of the Two Inequalities x y 0 1

Checking the Solution

To verify that the ordered pair (0, 1) satisfies both inequalities, we can plug it into both inequalities.

$y \leq -x + 1$ $1 \leq -0 + 1$ $1 \leq 1$ True
$y > x$ $1 > 0$ True

Since the ordered pair (0, 1) satisfies both inequalities, it is the solution to the system of inequalities.

Conclusion

In this article, we solved a system of inequalities and found the ordered pair that makes both inequalities true. We graphed the inequalities on a coordinate plane and found the intersection of the two shaded regions. We then checked the solution by plugging it into both inequalities. The ordered pair (0, 1) satisfies both inequalities, making it the solution to the system of inequalities.

Tips and Tricks

When solving a system of inequalities, it's essential to graph the inequalities on a coordinate plane to visualize the solution set.
The intersection of the two shaded regions represents the solution to the system of inequalities.
To verify the solution, plug it into both inequalities and check if it satisfies both conditions.

Real-World Applications

In economics, inequalities are used to model the relationship between variables such as supply and demand.
In engineering, inequalities are used to design and optimize systems such as electrical circuits and mechanical systems.
In computer science, inequalities are used to solve problems such as scheduling and resource allocation.

Common Mistakes

Failing to graph the inequalities on a coordinate plane can lead to incorrect solutions.
Not checking the solution by plugging it into both inequalities can result in incorrect answers.
Assuming that the intersection of the two shaded regions is the only solution can lead to incorrect conclusions.

Final Thoughts

Solving inequalities is a fundamental concept in mathematics that has numerous real-world applications. By graphing the inequalities on a coordinate plane and finding the intersection of the two shaded regions, we can find the ordered pair that makes both inequalities true. Remember to check the solution by plugging it into both inequalities to ensure accuracy. With practice and patience, you'll become proficient in solving inequalities and applying them to real-world problems.
Frequently Asked Questions: Solving Inequalities

In the previous article, we explored how to solve a system of inequalities and find the ordered pair that makes both inequalities true. In this article, we'll address some common questions and concerns that students and professionals may have when working with inequalities.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + by \leq c$ , where $a$ , $b$ , and $c$ are constants. A nonlinear inequality, on the other hand, is an inequality that cannot be written in this form. Examples of nonlinear inequalities include quadratic inequalities, such as $x^2 + 2x + 1 \leq 0$ .

Q: How do I graph a linear inequality on a coordinate plane?

A: To graph a linear inequality on a coordinate plane, follow these steps:

Graph the corresponding equation on the coordinate plane.
Determine the direction of the inequality symbol.
Shade the region that satisfies the inequality.

For example, to graph the inequality $y \leq -x + 1$ , graph the line $y = -x + 1$ and shade the region below the line.

Q: What is the intersection of two inequalities?

A: The intersection of two inequalities is the region that satisfies both inequalities. To find the intersection, graph both inequalities on a coordinate plane and shade the region that satisfies both inequalities.

Q: How do I check if a solution satisfies both inequalities?

A: To check if a solution satisfies both inequalities, plug the solution into both inequalities and check if it satisfies both conditions. For example, to check if the solution (0, 1) satisfies both inequalities, plug it into both inequalities:

$y \leq -x + 1$ $1 \leq -0 + 1$ $1 \leq 1$ True
$y > x$ $1 > 0$ True

Since the solution (0, 1) satisfies both inequalities, it is the solution to the system of inequalities.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

Failing to graph the inequalities on a coordinate plane
Not checking the solution by plugging it into both inequalities
Assuming that the intersection of the two shaded regions is the only solution

Q: How do I apply inequalities to real-world problems?

A: Inequalities have numerous real-world applications, including:

Modeling the relationship between variables such as supply and demand in economics
Designing and optimizing systems such as electrical circuits and mechanical systems in engineering
Solving problems such as scheduling and resource allocation in computer science

Q: What are some tips and tricks for solving inequalities?

A: Some tips and tricks for solving inequalities include:

Graphing the inequalities on a coordinate plane to visualize the solution set
Finding the intersection of the two shaded regions to find the solution
Checking the solution by plugging it into both inequalities to ensure accuracy

Q: What are some resources for learning more about inequalities?

A: Some resources for learning more about inequalities include:

Online tutorials and videos
Textbooks and reference books
Online forums and communities

Conclusion

In this article, we addressed some common questions and concerns that students and professionals may have when working with inequalities. We covered topics such as graphing linear inequalities, finding the intersection of two inequalities, and checking solutions. We also provided tips and tricks for solving inequalities and resources for learning more about inequalities. With practice and patience, you'll become proficient in solving inequalities and applying them to real-world problems.