Which Ordered Pair Makes Both Inequalities True?${ \begin{array}{l} y \ \textgreater \ -3x + 3 \ y \geq 2x - 2 \end{array} }$A. (1, 0) B. (-1, 1) C. (2, 2) D. (0, 3)

Mar 13, 2025 by ADMIN 172 views

**Solving Inequalities: Finding the Ordered Pair that Satisfies Both Conditions**

Introduction

In mathematics, inequalities are used to describe relationships between variables. When we have two inequalities, we need to find the values of the variables that satisfy both conditions. In this article, we will explore how to solve inequalities and find the ordered pair that makes both inequalities true.

Understanding the Inequalities

The two inequalities given are:

$y > -3x + 3$
$y \geq 2x - 2$

The first inequality states that $y$ is greater than $-3x + 3$ , while the second inequality states that $y$ is greater than or equal to $2x - 2$ .

Graphing the Inequalities

To visualize the inequalities, we can graph them on a coordinate plane. The first inequality, $y > -3x + 3$ , can be graphed as a line with a slope of $-3$ and a y-intercept of $3$ . The second inequality, $y \geq 2x - 2$ , can be graphed as a line with a slope of $2$ and a y-intercept of $-2$ .

Finding the Intersection

To find the ordered pair that satisfies both inequalities, we need to find the intersection of the two lines. The intersection point is where the two lines meet, and it represents the solution to the system of inequalities.

Solving the System of Inequalities

To solve the system of inequalities, we can use the following steps:

Graph the two inequalities on a coordinate plane.
Find the intersection point of the two lines.
Check if the intersection point satisfies both inequalities.

Step 1: Graph the Inequalities

To graph the inequalities, we can use the slope-intercept form of a line, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

For the first inequality, $y > -3x + 3$ , the slope is $-3$ and the y-intercept is $3$ . We can graph the line by plotting the y-intercept and then drawing a line with a slope of $-3$ .

For the second inequality, $y \geq 2x - 2$ , the slope is $2$ and the y-intercept is $-2$ . We can graph the line by plotting the y-intercept and then drawing a line with a slope of $2$ .

Step 2: Find the Intersection Point

To find the intersection point, we can set the two equations equal to each other and solve for $x$ and $y$ .

$-3x + 3 = 2x - 2$

Simplifying the equation, we get:

$-5x = -5$

Dividing both sides by $-5$ , we get:

$x = 1$

Substituting $x = 1$ into one of the original equations, we get:

$y = -3(1) + 3$

Simplifying the equation, we get:

$y = 0$

Therefore, the intersection point is $(1, 0)$ .

Step 3: Check the Intersection Point

To check if the intersection point satisfies both inequalities, we can substitute $x = 1$ and $y = 0$ into both inequalities.

For the first inequality, $y > -3x + 3$ , we get:

$0 > -3(1) + 3$

Simplifying the equation, we get:

$0 > 0$

This is true, so the intersection point satisfies the first inequality.

For the second inequality, $y \geq 2x - 2$ , we get:

$0 \geq 2(1) - 2$

Simplifying the equation, we get:

$0 \geq 0$

This is true, so the intersection point satisfies the second inequality.

Conclusion

In conclusion, the ordered pair that makes both inequalities true is $(1, 0)$ . This is the solution to the system of inequalities.

Answer

The correct answer is:

A. (1, 0)

Discussion

This problem requires the student to graph two inequalities on a coordinate plane, find the intersection point, and check if the intersection point satisfies both inequalities. The student must also be able to solve a system of linear equations and inequalities.

Tips and Variations

To make the problem more challenging, you can add more inequalities or change the slopes and y-intercepts of the lines.
To make the problem easier, you can use simpler inequalities or provide more guidance on how to graph the lines.
You can also ask the student to find the intersection point of the two lines using a different method, such as substitution or elimination.

Real-World Applications

This problem has real-world applications in fields such as economics, finance, and engineering. For example, in economics, the intersection point of two demand curves can represent the equilibrium price and quantity of a good. In finance, the intersection point of two investment portfolios can represent the optimal investment strategy. In engineering, the intersection point of two design constraints can represent the optimal design solution.

Conclusion

Q: What is the first step in solving a system of inequalities?

A: The first step in solving a system of inequalities is to graph the inequalities on a coordinate plane. This will help you visualize the relationships between the variables and find the intersection point.

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

A: To graph an inequality on a coordinate plane, you can use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept. You can plot the y-intercept and then draw a line with a slope of m.

Q: What is the intersection point of two lines?

A: The intersection point of two lines is the point where the two lines meet. This point represents the solution to the system of inequalities.

Q: How do I find the intersection point of two lines?

A: To find the intersection point of two lines, you can set the two equations equal to each other and solve for x and y. You can also use substitution or elimination to find the intersection point.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as > or <. A non-strict inequality is an inequality that is written with a non-strict symbol, such as ≥ or ≤.

Q: How do I check if the intersection point satisfies both inequalities?

A: To check if the intersection point satisfies both inequalities, you can substitute the x and y values of the intersection point into both inequalities and check if the inequality is true.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has many real-world applications in fields such as economics, finance, and engineering. For example, in economics, the intersection point of two demand curves can represent the equilibrium price and quantity of a good. In finance, the intersection point of two investment portfolios can represent the optimal investment strategy. In engineering, the intersection point of two design constraints can represent the optimal design solution.

Q: What are some tips for solving inequalities?

A: Here are some tips for solving inequalities:

Graph the inequalities on a coordinate plane to visualize the relationships between the variables.
Find the intersection point of the two lines.
Check if the intersection point satisfies both inequalities.
Use substitution or elimination to find the intersection point.
Be careful when working with strict and non-strict inequalities.

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

A: Here are some common mistakes to avoid when solving inequalities:

Failing to graph the inequalities on a coordinate plane.
Failing to find the intersection point of the two lines.
Failing to check if the intersection point satisfies both inequalities.
Making errors when substituting values into the inequalities.
Failing to consider the strict and non-strict inequalities.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems and exercises. You can also use online resources and practice tests to help you prepare for exams and assessments.

Conclusion

Solving inequalities and finding the ordered pair that makes both inequalities true is an important skill in mathematics. By following the steps outlined in this article and practicing regularly, you can become proficient in solving inequalities and apply this skill to real-world problems.