3. Which Ordered Pair Is A Solution To The System Of Inequalities $\left\{\begin{array}{l}y \leq |x| \\ Y \ \textless \ -2x - 1\end{array}\right.$?A. $(0, 4$\]B. $(2, -1$\]C. $(3, 1$\]D. $(-3, 1$\]

Introduction

In mathematics, solving systems of inequalities is a crucial concept that involves finding the solution set of a system of linear or nonlinear inequalities. In this article, we will focus on solving a system of inequalities involving absolute value and linear inequalities. We will use the given system of inequalities $\left\{\begin{array}{l}y \leq |x| \\ y \ \textless \ -2x - 1\end{array}\right.$ to illustrate the step-by-step process of solving systems of inequalities.

Understanding Absolute Value Inequalities

Before we dive into solving the system of inequalities, let's first understand how to solve absolute value inequalities. An absolute value inequality is of the form $|x| \leq a$, where $a$ is a positive real number. To solve this inequality, we need to consider two cases:

• Case 1: $x \geq 0$
• Case 2: $x < 0$

For case 1, we have $x \leq a$, and for case 2, we have $-x \leq a$, which simplifies to $x \geq -a$.

Solving the First Inequality

The first inequality in the system is $y \leq |x|$. To solve this inequality, we need to consider the two cases:

• Case 1: $x \geq 0$
• Case 2: $x < 0$

For case 1, we have $y \leq x$, and for case 2, we have $y \leq -x$.

Solving the Second Inequality

The second inequality in the system is $y < -2x - 1$. To solve this inequality, we need to isolate $y$ by adding $2x + 1$ to both sides of the inequality. This gives us $y + 2x + 1 < 0$.

Graphing the Inequalities

To visualize the solution set of the system of inequalities, we need to graph the two inequalities on a coordinate plane. The first inequality $y \leq |x|$ can be graphed as a V-shaped graph, with the vertex at the origin $(0, 0)$. The second inequality $y < -2x - 1$ can be graphed as a line with a slope of $-2$ and a y-intercept of $-1$.

Finding the Solution Set

To find the solution set of the system of inequalities, we need to find the region where the two inequalities overlap. We can do this by graphing the two inequalities on a coordinate plane and identifying the region where both inequalities are satisfied.

Analyzing the Answer Choices

Now that we have graphed the two inequalities, let's analyze the answer choices to see which one satisfies both inequalities.

• A. $(0, 4)$: This point lies above the line $y = -2x - 1$, so it does not satisfy the second inequality.
• B. $(2, -1)$: This point lies below the line $y = -2x - 1$, so it satisfies the second inequality. However, it lies above the V-shaped graph of the first inequality, so it does not satisfy the first inequality.
• C. $(3, 1)$: This point lies above the line $y = -2x - 1$, so it does not satisfy the second inequality.
• D. $(-3, 1)$: This point lies below the line $y = -2x - 1$, so it satisfies the second inequality. It also lies below the V-shaped graph of the first inequality, so it satisfies the first inequality.

Conclusion

In conclusion, the correct answer is D. $(-3, 1)$. This point satisfies both inequalities and lies in the region where the two inequalities overlap.

Final Answer

The final answer is $\boxed{D}$.

Additional Tips and Resources

• To solve systems of inequalities, it's essential to understand how to solve absolute value inequalities and linear inequalities.
• Graphing the inequalities on a coordinate plane can help visualize the solution set.
• Analyzing the answer choices can help identify the correct solution.
• For more practice problems and resources, check out the following websites:
  • Khan Academy: Systems of Inequalities
  • Mathway: Systems of Inequalities
  • IXL: Systems of Inequalities
    Frequently Asked Questions: Systems of Inequalities =====================================================

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are combined to form a single system. Each inequality in the system is a statement that describes a relationship between two or more variables.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the solution set of each inequality and then identify the region where all the inequalities overlap. You can use graphing, substitution, or elimination methods to solve the inequalities.

Q: What is the difference between a system of linear inequalities and a system of nonlinear inequalities?

A: A system of linear inequalities consists of two or more linear inequalities, while a system of nonlinear inequalities consists of two or more nonlinear inequalities. Nonlinear inequalities can be quadratic, polynomial, or absolute value inequalities.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph each inequality separately and then identify the region where all the inequalities overlap. You can use a coordinate plane to graph the inequalities.

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

A: The solution set of a system of inequalities is the region where all the inequalities overlap. It is the set of all points that satisfy all the inequalities in the system.

Q: How do I find the solution set of a system of inequalities?

A: To find the solution set of a system of inequalities, you need to graph each inequality separately and then identify the region where all the inequalities overlap. You can use graphing, substitution, or elimination methods to solve the 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 identifying the correct solution set
• Not considering all possible cases
• Not using the correct method to solve the inequalities

Q: How do I check my answer when solving a system of inequalities?

A: To check your answer when solving a system of inequalities, you need to substitute the solution into each inequality and verify that it satisfies all the inequalities. You can also graph the solution on a coordinate plane to verify that it lies in the correct region.

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

A: Systems of inequalities have many real-world applications, including:

• Optimization problems
• Resource allocation problems
• Scheduling problems
• Budgeting problems

Q: How do I use technology to solve systems of inequalities?

A: You can use technology such as graphing calculators or computer software to solve systems of inequalities. These tools can help you graph the inequalities, identify the solution set, and verify your answer.

Q: What are some additional resources for learning about systems of inequalities?

A: Some additional resources for learning about systems of inequalities include:

• Khan Academy: Systems of Inequalities
• Mathway: Systems of Inequalities
• IXL: Systems of Inequalities
• Online textbooks and tutorials
• Video lectures and tutorials