Solve The Inequality: X + 5 Y \textless − 30 X + 5y \ \textless \ -30 X + 5 Y \textless − 30

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Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving the inequality $x + 5y < -30$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and graphical representations to help you understand the concept.

Understanding the Inequality

The given inequality is $x + 5y < -30$. This means that the sum of $x$ and $5y$ must be less than $-30$. To solve this inequality, we need to isolate the variables $x$ and $y$.

Isolating the Variables

To isolate the variables, we can start by subtracting $5y$ from both sides of the inequality. This gives us:

$x < -30 - 5y$

Simplifying the Inequality

Now, we can simplify the inequality by combining the constants on the right-hand side:

$x < -30 - 5y$

Graphical Representation

To visualize the solution, we can graph the inequality on a coordinate plane. The inequality $x < -30 - 5y$ represents a region in the plane where the sum of $x$ and $5y$ is less than $-30$.

Slope-Intercept Form

To graph the inequality, we can rewrite it in slope-intercept form, which is $y = mx + b$. In this case, the slope-intercept form of the inequality is:

$y = -\frac{1}{5}x - 6$

Graphing the Inequality

To graph the inequality, we can plot the line $y = -\frac{1}{5}x - 6$ and shade the region below it. The shaded region represents the solution to the inequality.

Solution to the Inequality

The solution to the inequality $x + 5y < -30$ is the region below the line $y = -\frac{1}{5}x - 6$. This region includes all the points $(x, y)$ that satisfy the inequality.

Conclusion

Solving linear inequalities is an essential skill in mathematics, and the inequality $x + 5y < -30$ is a great example of how to apply these skills. By following the steps outlined in this article, you should be able to solve the inequality and understand the concept of linear inequalities.

Frequently Asked Questions

• What is a linear inequality? A linear inequality is an inequality that can be written in the form $ax + by < c$, where $a$, $b$, and $c$ are constants.
• How do I solve a linear inequality? To solve a linear inequality, you can follow the steps outlined in this article, which include isolating the variables, simplifying the inequality, and graphing the solution.
• What is the solution to the inequality $x + 5y < -30$? The solution to the inequality $x + 5y < -30$ is the region below the line $y = -\frac{1}{5}x - 6$.

Additional Resources

• Linear Inequalities Tutorial: This tutorial provides a comprehensive overview of linear inequalities, including how to solve them and graph their solutions.
• Graphing Linear Inequalities: This article provides a step-by-step guide to graphing linear inequalities, including how to plot the line and shade the region.
• Solving Linear Inequalities: This article provides a detailed explanation of how to solve linear inequalities, including how to isolate the variables and simplify the inequality.
  

Introduction

Solving linear inequalities can be a challenging task, but with the right guidance, it can become a breeze. In this article, we will address some of the most frequently asked questions about solving linear inequalities, including how to isolate the variables, simplify the inequality, and graph the solution.

Q&A: Solving Linear Inequalities

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + by < c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can follow these steps:

1. Isolate the variables by adding or subtracting the same value to both sides of the inequality.
2. Simplify the inequality by combining like terms.
3. Graph the solution by plotting the line and shading the region.

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

A: A linear equation is an equation that can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants. A linear inequality, on the other hand, is an inequality that can be written in the form $ax + by < c$, where $a$, $b$, and $c$ are constants.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you can follow these steps:

1. Plot the line by finding the x-intercept and y-intercept.
2. Shade the region below the line for a less-than inequality.
3. Shade the region above the line for a greater-than inequality.

Q: What is the solution to the inequality $x + 5y < -30$?

A: The solution to the inequality $x + 5y < -30$ is the region below the line $y = -\frac{1}{5}x - 6$.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, you can follow these steps:

1. Look at the sign of the coefficient of the variable.
2. If the coefficient is positive, the inequality is greater-than.
3. If the coefficient is negative, the inequality is less-than.

Q: Can I use a calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, it's always a good idea to check your work by graphing the solution.

Q: What is the importance of solving linear inequalities?

A: Solving linear inequalities is an essential skill in mathematics, as it helps you to understand the concept of linear inequalities and how to apply them in real-world problems.

Conclusion

Solving linear inequalities can be a challenging task, but with the right guidance, it can become a breeze. By following the steps outlined in this article, you should be able to solve linear inequalities and understand the concept of linear inequalities.

Additional Resources

• Linear Inequalities Tutorial: This tutorial provides a comprehensive overview of linear inequalities, including how to solve them and graph their solutions.
• Graphing Linear Inequalities: This article provides a step-by-step guide to graphing linear inequalities, including how to plot the line and shade the region.
• Solving Linear Inequalities: This article provides a detailed explanation of how to solve linear inequalities, including how to isolate the variables and simplify the inequality.

Frequently Asked Questions: Solving Linear Inequalities

• What is a linear inequality? A linear inequality is an inequality that can be written in the form $ax + by < c$, where $a$, $b$, and $c$ are constants.
• How do I solve a linear inequality? To solve a linear inequality, you can follow the steps outlined in this article, which include isolating the variables, simplifying the inequality, and graphing the solution.
• What is the solution to the inequality $x + 5y < -30$? The solution to the inequality $x + 5y < -30$ is the region below the line $y = -\frac{1}{5}x - 6$.

Related Articles

• Linear Equations: This article provides a comprehensive overview of linear equations, including how to solve them and graph their solutions.
• Graphing Linear Equations: This article provides a step-by-step guide to graphing linear equations, including how to plot the line and find the x-intercept and y-intercept.
• Solving Systems of Linear Equations: This article provides a detailed explanation of how to solve systems of linear equations, including how to use substitution and elimination methods.