Amber Is Solving The Inequality ∣ X + 6 ∣ − 12 \textless 13 |x+6|-12\ \textless \ 13 ∣ X + 6∣ − 12 \textless 13 By Graphing. Which Equations Should Amber Graph?A. Y 1 = ∣ X + 6 ∣ , Y 2 = 25 Y_1=|x+6|, Y_2=25 Y 1 = ∣ X + 6∣ , Y 2 = 25 B. Y 1 = X + 6 , Y 2 = 25 Y_1=x+6, Y_2=25 Y 1 = X + 6 , Y 2 = 25 C. Y 1 = ∣ X + 6 ∣ , Y 2 = 13 Y_1=|x+6|, Y_2=13 Y 1 = ∣ X + 6∣ , Y 2 = 13 D. Y 1 = X + 6 , Y 2 = 13 Y_1=x+6, Y_2=13 Y 1 = X + 6 , Y 2 = 13

Mar 13, 2025 by ADMIN 455 views

**Solving Inequalities with Graphing: A Step-by-Step Guide**

Introduction

Graphing is a powerful tool for solving inequalities, allowing us to visualize the solution set and make it easier to understand. In this article, we will explore how to solve the inequality $|x+6|-12\ \textless \ 13$ by graphing, and determine which equations Amber should graph.

Understanding the Inequality

Before we dive into graphing, let's take a closer look at the inequality $|x+6|-12\ \textless \ 13$ . This inequality involves absolute value, which means we need to consider both the positive and negative cases.

The absolute value of a number $a$ is defined as $|a| = a$ if $a \geq 0$ , and $|a| = -a$ if $a < 0$ . In this case, we have $|x+6|$ , which means we need to consider both the positive and negative values of $x+6$ .

Graphing the Inequality

To graph the inequality, we need to graph two equations: $y_1 = |x+6|$ and $y_2 = 25$ . The first equation represents the absolute value function, while the second equation represents a horizontal line.

Graphing the Absolute Value Function

The absolute value function $y_1 = |x+6|$ has a minimum value of 0 at $x = -6$ , and increases linearly as $x$ increases. The graph of this function is a V-shaped graph, with the vertex at $(-6, 0)$ .

Graphing the Horizontal Line

The horizontal line $y_2 = 25$ is a straight line that intersects the y-axis at $y = 25$ . This line represents the upper bound of the inequality.

Determining the Solution Set

To determine the solution set, we need to find the region where the absolute value function is less than the horizontal line. This means we need to find the values of $x$ where $|x+6| < 25$ .

Solving the Inequality

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

Graph the absolute value function: Graph the absolute value function $y_1 = |x+6|$ .
Graph the horizontal line: Graph the horizontal line $y_2 = 25$ .
Find the intersection points: Find the intersection points of the absolute value function and the horizontal line.
Determine the solution set: Determine the region where the absolute value function is less than the horizontal line.

Which Equations Should Amber Graph?

Based on the steps above, Amber should graph the following equations:

$y_1 = |x+6|$
$y_2 = 13$

The correct answer is C. $y_1=|x+6|, y_2=13$ .

Conclusion

Introduction

In our previous article, we explored how to solve the inequality $|x+6|-12\ \textless \ 13$ by graphing. We determined that Amber should graph the equations $y_1 = |x+6|$ and $y_2 = 13$ . In this article, we will answer some common questions about solving inequalities with graphing.

Q&A

Q: What is the purpose of graphing in solving inequalities?

A: The purpose of graphing in solving inequalities is to visualize the solution set and make it easier to understand. By graphing the absolute value function and the horizontal line, we can determine the region where the absolute value function is less than the horizontal line.

Q: How do I graph the absolute value function?

A: To graph the absolute value function, you can use the following steps:

Graph the V-shaped graph: Graph a V-shaped graph with the vertex at $(-6, 0)$ .
Label the x-axis: Label the x-axis with the values of $x$ .
Label the y-axis: Label the y-axis with the values of $y$ .

Q: How do I graph the horizontal line?

A: To graph the horizontal line, you can use the following steps:

Draw a horizontal line: Draw a horizontal line that intersects the y-axis at $y = 25$ .
Label the x-axis: Label the x-axis with the values of $x$ .
Label the y-axis: Label the y-axis with the values of $y$ .

Q: How do I determine the solution set?

A: To determine the solution set, you can use the following steps:

Find the intersection points: Find the intersection points of the absolute value function and the horizontal line.
Determine the region: Determine the region where the absolute value function is less than the horizontal line.

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

A: Some common mistakes to avoid when graphing inequalities include:

Graphing the wrong equation: Make sure to graph the correct equation, such as $y_1 = |x+6|$ and $y_2 = 13$ .
Not labeling the axes: Make sure to label the x-axis and y-axis with the correct values.
Not determining the solution set: Make sure to determine the solution set by finding the intersection points and determining the region.

Q: Can I use graphing to solve other types of inequalities?

A: Yes, you can use graphing to solve other types of inequalities, such as linear inequalities and quadratic inequalities.

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

A: Some real-world applications of solving inequalities with graphing include:

Optimization problems: Graphing can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
Data analysis: Graphing can be used to analyze data and determine the solution set.
Engineering: Graphing can be used to solve engineering problems, such as designing a system or optimizing a process.

Conclusion

Solving inequalities with graphing is a powerful tool that allows us to visualize the solution set and make it easier to understand. By graphing the absolute value function and the horizontal line, we can determine the solution set and find the values of $x$ that satisfy the inequality. In this article, we answered some common questions about solving inequalities with graphing, and provided some tips and tricks for graphing inequalities.