Solve The Following Inequality Algebraically. ∣ X − 9 ∣ ≤ 2 |x-9| \leq 2 ∣ X − 9∣ ≤ 2

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Introduction

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Absolute value inequalities are a fundamental concept in algebra, and solving them requires a clear understanding of the properties of absolute value functions. In this article, we will focus on solving the inequality $|x-9| \leq 2$ algebraically. We will break down the solution into manageable steps, using a combination of mathematical reasoning and visual aids to illustrate the process.

Understanding Absolute Value Functions

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Before we dive into solving the inequality, let's take a moment to understand the properties of absolute value functions. The absolute value function, denoted by $|x|$, is defined as:

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

This function takes any real number $x$ and returns its distance from zero on the number line. In other words, it returns the magnitude of $x$ without considering its sign.

Solving the Inequality

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Now that we have a solid understanding of absolute value functions, let's tackle the inequality $|x-9| \leq 2$. To solve this inequality, we need to consider two cases:

Case 1: $x-9 \geq 0$

In this case, the absolute value function simplifies to $|x-9| = x-9$. We can then rewrite the inequality as:

$$x-9 \leq 2$$

To solve for $x$, we add 9 to both sides of the inequality:

$$x \leq 11$$

So, in this case, the solution is $x \leq 11$.

Case 2: $x-9 < 0$

In this case, the absolute value function simplifies to $|x-9| = -(x-9)$. We can then rewrite the inequality as:

$$-(x-9) \leq 2$$

To solve for $x$, we multiply both sides of the inequality by -1, which reverses the direction of the inequality:

$$x-9 \geq -2$$

We then add 9 to both sides of the inequality:

$$x \geq 7$$

So, in this case, the solution is $x \geq 7$.

Combining the Solutions

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Now that we have solved the inequality for both cases, we need to combine the solutions to get the final answer. We can do this by taking the union of the two solutions:

$$x \leq 11 \text{ or } x \geq 7$$

This can be rewritten as a single interval:

$$7 \leq x \leq 11$$

Conclusion

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Solving absolute value inequalities requires a clear understanding of the properties of absolute value functions. By breaking down the solution into manageable steps and using a combination of mathematical reasoning and visual aids, we can arrive at the final answer. In this article, we solved the inequality $|x-9| \leq 2$ algebraically, using two cases to consider the different signs of the expression inside the absolute value function. The final solution is $7 \leq x \leq 11$.

Visualizing the Solution

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To help illustrate the solution, let's visualize the graph of the absolute value function $|x-9|$ and the inequality $|x-9| \leq 2$. We can do this by plotting the graph of the absolute value function and shading the region that satisfies the inequality.

import numpy as np
import matplotlib.pyplot as plt
def abs_value(x):
return np.abs(x-9)

x = np.linspace(-10, 20, 400)

y = abs_value(x)

plt.plot(x, y)
plt.fill_between(x, 0, 2, color='blue', alpha=0.3)
plt.fill_between(x, -2, 0, color='blue', alpha=0.3)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of |x-9| and the Inequality |x-9| ≤ 2')
plt.show()

This code generates a graph of the absolute value function and shades the region that satisfies the inequality. The resulting graph shows the solution to the inequality $|x-9| \leq 2$.

Applications of Absolute Value Inequalities

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Absolute value inequalities have numerous applications in mathematics and real-world problems. Some examples include:

• Distance and Time: In physics, absolute value inequalities can be used to model the distance and time traveled by an object.
• Finance: In finance, absolute value inequalities can be used to model the value of assets and investments.
• Computer Science: In computer science, absolute value inequalities can be used to model the complexity of algorithms and data structures.

Conclusion

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In conclusion, solving absolute value inequalities requires a clear understanding of the properties of absolute value functions. By breaking down the solution into manageable steps and using a combination of mathematical reasoning and visual aids, we can arrive at the final answer. The final solution to the inequality $|x-9| \leq 2$ is $7 \leq x \leq 11$.

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Introduction

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In our previous article, we explored the concept of absolute value inequalities and solved the inequality $|x-9| \leq 2$ algebraically. However, we know that there are many more questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions about absolute value inequalities.

Q: What is an absolute value inequality?

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A: An absolute value inequality is an inequality that involves the absolute value function. It is a mathematical statement that compares the absolute value of an expression to a constant or another expression.

Q: How do I solve an absolute value inequality?

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A: To solve an absolute value inequality, you need to consider two cases:

• Case 1: The expression inside the absolute value function is non-negative.
• Case 2: The expression inside the absolute value function is negative.

You then solve each case separately and combine the solutions to get the final answer.

Q: What is the difference between an absolute value inequality and a linear inequality?

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A: An absolute value inequality is a type of inequality that involves the absolute value function, while a linear inequality is a type of inequality that involves a linear expression. For example, the inequality $|x-9| \leq 2$ is an absolute value inequality, while the inequality $x \leq 11$ is a linear inequality.

Q: Can I use the same method to solve all absolute value inequalities?

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A: No, you cannot use the same method to solve all absolute value inequalities. The method you use depends on the specific inequality you are solving. For example, if the inequality involves a quadratic expression inside the absolute value function, you may need to use a different method to solve it.

Q: How do I graph an absolute value inequality?

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A: To graph an absolute value inequality, you need to graph the absolute value function and shade the region that satisfies the inequality. You can use a graphing calculator or software to help you graph the function and shade the region.

Q: Can I use absolute value inequalities to model real-world problems?

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A: Yes, you can use absolute value inequalities to model real-world problems. For example, you can use absolute value inequalities to model the distance and time traveled by an object, or the value of assets and investments in finance.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

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A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not considering both cases when solving the inequality.
• Not combining the solutions correctly.
• Not checking the solutions to make sure they satisfy the original inequality.

Q: How do I check my solutions to an absolute value inequality?

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A: To check your solutions to an absolute value inequality, you need to plug each solution back into the original inequality and make sure it is true. You can also use a graphing calculator or software to help you check your solutions.

Q: Can I use absolute value inequalities to solve systems of equations?

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A: Yes, you can use absolute value inequalities to solve systems of equations. For example, you can use absolute value inequalities to solve systems of linear equations or quadratic equations.

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

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A: Some real-world applications of absolute value inequalities include:

• Distance and Time: In physics, absolute value inequalities can be used to model the distance and time traveled by an object.
• Finance: In finance, absolute value inequalities can be used to model the value of assets and investments.
• Computer Science: In computer science, absolute value inequalities can be used to model the complexity of algorithms and data structures.

Conclusion

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In conclusion, absolute value inequalities are a powerful tool for solving mathematical problems and modeling real-world situations. By understanding the properties of absolute value functions and how to solve absolute value inequalities, you can apply these concepts to a wide range of problems and applications. We hope this Q&A article has helped to clarify any doubts you may have had about absolute value inequalities.