Solve The System Of Inequalities:${ \begin{array}{l} 2x - 3y \leq 12 \ y \ \textless \ -3 \end{array} }$

Mar 13, 2025 by ADMIN 108 views

Solving Systems of Inequalities: A Comprehensive Guide

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Introduction

Solving systems of inequalities is a crucial concept in mathematics, particularly in algebra and calculus. It involves finding the solution set that satisfies multiple inequalities simultaneously. In this article, we will delve into the world of systems of inequalities, focusing on the given system: $2x - 3y \leq 12$ and $y < -3$ . We will explore the steps to solve this system, providing a clear and concise explanation of the process.

Understanding the Basics

Before diving into the solution, it's essential to understand the basics of inequalities. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In the given system, we have two inequalities:

$2x - 3y \leq 12$
$y < -3$

The first inequality is a linear inequality, while the second is a strict inequality. We will use these inequalities to find the solution set.

Solving the First Inequality

To solve the first inequality, $2x - 3y \leq 12$ , we can use the following steps:

Step 1: Isolate the Variable

We can isolate the variable $y$ by subtracting $2x$ from both sides of the inequality:

-3y \leq 12 - 2x

Step 2: Divide by the Coefficient

Next, we divide both sides of the inequality by $-3$ :

y \geq \frac{12 - 2x}{-3}

Step 3: Simplify the Expression

Simplifying the expression, we get:

y \leq -\frac{2}{3}x + 4

This is the solution to the first inequality.

Solving the Second Inequality

The second inequality, $y < -3$ , is a strict inequality. We can solve this inequality by simply stating that $y$ is less than $-3$ .

Combining the Inequalities

Now that we have solved both inequalities, we can combine them to find the solution set. We will use the following steps:

Step 1: Graph the Inequalities

To visualize the solution set, we can graph the inequalities on a coordinate plane. The first inequality, $y \leq -\frac{2}{3}x + 4$ , is a line with a slope of $-\frac{2}{3}$ and a y-intercept of $4$ . The second inequality, $y < -3$ , is a horizontal line at $y = -3$ .

Step 2: Find the Intersection

The solution set is the region where the two inequalities intersect. We can find the intersection by solving the system of equations:

y = -\frac{2}{3}x + 4

y = -3

Substituting $y = -3$ into the first equation, we get:

-3 = -\frac{2}{3}x + 4

Solving for $x$ , we get:

x = 9

Substituting $x = 9$ into the first equation, we get:

y = -\frac{2}{3}(9) + 4

Simplifying the expression, we get:

y = -6 + 4

y = -2

Therefore, the intersection point is $(9, -2)$ .

Step 3: Shade the Region

The solution set is the region below the line $y = -\frac{2}{3}x + 4$ and above the line $y = -3$ . We can shade this region to visualize the solution set.

Conclusion

In conclusion, solving systems of inequalities involves finding the solution set that satisfies multiple inequalities simultaneously. We have solved the given system of inequalities, $2x - 3y \leq 12$ and $y < -3$ , using the steps outlined above. The solution set is the region below the line $y = -\frac{2}{3}x + 4$ and above the line $y = -3$ . We can use this solution set to make informed decisions in various real-world applications.

Final Answer

The final answer is the solution set, which is the region below the line $y = -\frac{2}{3}x + 4$ and above the line $y = -3$ .

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Q&A: Frequently Asked Questions

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are solved simultaneously to find the solution set.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you can use the following steps:

Graph the inequalities on a coordinate plane.
Find the intersection of the inequalities.
Shade the region that satisfies both inequalities.

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

A: A linear inequality is an inequality that can be written in the form $ax + by \leq c$ or $ax + by \geq c$ , where $a$ , $b$ , and $c$ are constants. A strict inequality is an inequality that is written with a strict symbol, such as $y < -3$ .

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you can use the following steps:

Graph the equation of the line.
Test a point in each region to determine which region satisfies the inequality.

Q: What is the solution set of a system of inequalities?

A: The solution set of a system of inequalities is the region that satisfies both inequalities.

Q: How do I find the intersection of two inequalities?

A: To find the intersection of two inequalities, you can solve the system of equations formed by the two inequalities.

Q: What is the importance of solving systems of inequalities?

A: Solving systems of inequalities is important in various real-world applications, such as:

Finance: Solving systems of inequalities can help you determine the best investment strategy.
Business: Solving systems of inequalities can help you determine the best pricing strategy.
Science: Solving systems of inequalities can help you determine the best experimental design.

Q: Can I use technology to solve systems of inequalities?

A: Yes, you can use technology, such as graphing calculators or computer software, to solve systems of inequalities.

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

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

Not graphing the inequalities correctly.
Not finding the intersection of the inequalities.
Not shading the region correctly.

Q: How can I practice solving systems of inequalities?

A: You can practice solving systems of inequalities by:

Working through examples in a textbook or online resource.
Creating your own examples and solving them.
Using technology to solve systems of inequalities.

Additional Resources

Khan Academy: Systems of Inequalities
Mathway: Systems of Inequalities
Wolfram Alpha: Systems of Inequalities

Conclusion

Solving systems of inequalities is an important concept in mathematics that has various real-world applications. By following the steps outlined in this article, you can solve systems of inequalities and find the solution set. Remember to practice solving systems of inequalities to become proficient in this concept.

Final Answer

The final answer is the solution set of the system of inequalities, which is the region that satisfies both inequalities.