Solve The Inequality:$\[ 2x + Y \leq -5 \\]

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 $2x + y \leq -5$. We will break down the solution into manageable steps, using clear and concise language to ensure that readers understand the process.

Understanding the Inequality

Before we dive into solving the inequality, let's take a closer look at what it represents. The inequality $2x + y \leq -5$ is a linear inequality, which means that it can be represented graphically as a line on a coordinate plane. The inequality is in the form of $ax + by \leq c$, where $a$, $b$, and $c$ are constants.

The Components of the Inequality

To solve the inequality, we need to understand the components that make it up. The inequality has two variables, $x$ and $y$, and a constant term, $-5$. The coefficient of $x$ is $2$, and the coefficient of $y$ is $1$. The inequality is also subject to the condition that the result must be less than or equal to $-5$.

Solving the Inequality

To solve the inequality, we can use the following steps:

Step 1: Isolate the Variable

The first step in solving the inequality is to isolate the variable $x$. We can do this by subtracting $y$ from both sides of the inequality.

${ 2x + y \leq -5 }$

Subtracting $y$ from both sides gives us:

${ 2x \leq -5 - y }$

Step 2: Divide by the Coefficient

Next, we need to divide both sides of the inequality by the coefficient of $x$, which is $2$.

${ 2x \leq -5 - y }$

Dividing both sides by $2$ gives us:

${ x \leq \frac{-5 - y}{2} }$

Step 3: Write the Solution in Interval Notation

The final step in solving the inequality is to write the solution in interval notation. The solution is all values of $x$ that satisfy the inequality.

${ x \leq \frac{-5 - y}{2} }$

This can be written in interval notation as:

${ x \in \left[ \frac{-5 - y}{2}, \infty \right) }$

Graphing the Solution

To visualize the solution, we can graph the inequality on a coordinate plane. The inequality is a line with a slope of $-1/2$ and a y-intercept of $-5/2$. The solution is all points on or below this line.

Conclusion

Solving linear inequalities is an essential skill for students to master. By following the steps outlined in this article, readers can solve the inequality $2x + y \leq -5$. The solution is all values of $x$ that satisfy the inequality, and it can be written in interval notation as $x \in \left[ \frac{-5 - y}{2}, \infty \right)$. By practicing solving linear inequalities, readers can develop a deeper understanding of the subject and improve their problem-solving skills.

Additional Resources

For readers who want to learn more about solving linear inequalities, here are some additional resources:

• Khan Academy: Linear Inequalities
• Mathway: Solving Linear Inequalities
• Wolfram Alpha: Linear Inequalities

Frequently Asked Questions

Q: What is a linear inequality? A: A linear inequality is an inequality that can be represented graphically as a line on a coordinate plane.

Q: How do I solve a linear inequality? A: To solve a linear inequality, you need to isolate the variable, divide by the coefficient, and write the solution in interval notation.

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Introduction

In our previous article, we discussed how to solve the inequality $2x + y \leq -5$. We broke down the solution into manageable steps and provided a clear and concise explanation of the process. In this article, we will continue to explore the topic of solving linear inequalities by answering some of the most frequently asked questions.

Q&A

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be represented graphically as a line on a coordinate plane. It is an inequality that involves a linear expression and a constant term.

Q: How do I solve a linear inequality?

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

1. Isolate the variable by adding or subtracting the same value to both sides of the inequality.
2. Divide both sides of the inequality by the coefficient of the variable.
3. Write the solution in interval notation.

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

A: A linear equation is an equation that can be represented graphically as a line on a coordinate plane. It is an equation that involves a linear expression and a constant term. A linear inequality, on the other hand, is an inequality that can be represented graphically as a line on a coordinate plane. It is an inequality that involves a linear expression and a constant term.

Q: How do I graph a linear inequality?

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

1. Graph the corresponding linear equation.
2. Determine the direction of the inequality.
3. Shade the region that satisfies the inequality.

Q: What is the solution to the inequality $2x + y \leq -5$?

A: The solution to the inequality $2x + y \leq -5$ is all values of $x$ that satisfy the inequality. The solution can be written in interval notation as $x \in \left[ \frac{-5 - y}{2}, \infty \right)$.

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

A: To determine the direction of the inequality, you need to look at the sign of the coefficient of the variable. If the coefficient is positive, the inequality is pointing upwards. If the coefficient is negative, the inequality is pointing downwards.

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

A: A strict inequality is an inequality that is represented by a strict symbol, such as $<$ or $>$. A non-strict inequality is an inequality that is represented by a non-strict symbol, such as $\leq$ or $\geq$.

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

A: To solve a system of linear inequalities, you need to follow these steps:

1. Graph each inequality on a coordinate plane.
2. Determine the region that satisfies all the inequalities.
3. Write the solution in interval notation.

Conclusion

Solving linear inequalities is an essential skill for students to master. By following the steps outlined in this article, readers can solve linear inequalities and develop a deeper understanding of the subject. We hope that this Q&A guide has been helpful in answering some of the most frequently asked questions about solving linear inequalities.

Additional Resources

For readers who want to learn more about solving linear inequalities, here are some additional resources:

• Khan Academy: Linear Inequalities
• Mathway: Solving Linear Inequalities
• Wolfram Alpha: Linear Inequalities

Frequently Asked Questions

Q: What is a linear inequality? A: A linear inequality is an inequality that can be represented graphically as a line on a coordinate plane.

Q: How do I solve a linear inequality? A: To solve a linear inequality, you need to isolate the variable, divide by the coefficient, and write the solution in interval notation.

Q: What is the solution to the inequality $2x + y \leq -5$? A: The solution to the inequality $2x + y \leq -5$ is all values of $x$ that satisfy the inequality, which can be written in interval notation as $x \in \left[ \frac{-5 - y}{2}, \infty \right)$.

Q: How do I determine the direction of the inequality? A: To determine the direction of the inequality, you need to look at the sign of the coefficient of the variable. If the coefficient is positive, the inequality is pointing upwards. If the coefficient is negative, the inequality is pointing downwards.

Q: What is the difference between a strict inequality and a non-strict inequality? A: A strict inequality is an inequality that is represented by a strict symbol, such as $<$ or $>$. A non-strict inequality is an inequality that is represented by a non-strict symbol, such as $\leq$ or $\geq$.

Q: How do I solve a system of linear inequalities? A: To solve a system of linear inequalities, you need to graph each inequality on a coordinate plane, determine the region that satisfies all the inequalities, and write the solution in interval notation.