Fuaad Solved An Absolute Value Inequality And Expressed The Solution As $-12 \ \textless \ X \ \textless \ 7$. Which Is Fuaad's Solution?A. $(-\infty, -12) \cup (7, \infty$\]B. $(-12, 7\]C. $\{x \mid -12 \ \textless \

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Understanding Absolute Value Inequalities

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Absolute value inequalities involve the absolute value of a variable or expression, and they can be solved using various methods. In this case, Fuaad has solved an absolute value inequality and expressed the solution as $-12 \ \textless \ x \ \textless \ 7$. To determine Fuaad's solution, we need to understand the concept of absolute value inequalities and how to express their solutions.

Absolute Value Inequality Solutions

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An absolute value inequality is of the form $|x| \ \textless \ k$, where $k$ is a positive constant. The solution to this inequality is all values of $x$ that satisfy the inequality. In the case of Fuaad's solution, we have $-12 \ \textless \ x \ \textless \ 7$. This means that $x$ is greater than $-12$ and less than $7$.

Expressing Solutions as Intervals

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To express the solution as an interval, we need to use the notation $(-\infty, a) \cup (b, \infty)$, where $a$ and $b$ are the endpoints of the interval. In this case, the endpoints are $-12$ and $7$. However, we need to be careful when expressing the solution as an interval, as the endpoints are not included in the solution.

Fuaad's Solution

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Fuaad's solution is expressed as $-12 \ \textless \ x \ \textless \ 7$. This means that $x$ is greater than $-12$ and less than $7$. To express this solution as an interval, we need to use the notation $(-\infty, -12) \cup (7, \infty)$.

Conclusion

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In conclusion, Fuaad's solution to the absolute value inequality is $(-\infty, -12) \cup (7, \infty)$. This solution is expressed as an interval, where the endpoints are $-12$ and $7$. The solution is all values of $x$ that satisfy the inequality $-12 \ \textless \ x \ \textless \ 7$.

Step-by-Step Solution

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To solve the absolute value inequality, we need to follow these steps:

1. Write the inequality in the form $|x| \ \textless \ k$.
2. Determine the value of $k$.
3. Express the solution as an interval using the notation $(-\infty, a) \cup (b, \infty)$.

Example

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Let's consider an example to illustrate the solution to an absolute value inequality. Suppose we have the inequality $|x| \ \textless \ 3$. To solve this inequality, we need to follow the steps outlined above.

1. Write the inequality in the form $|x| \ \textless \ k$. In this case, $k = 3$.
2. Determine the value of $k$. In this case, $k = 3$.
3. Express the solution as an interval using the notation $(-\infty, a) \cup (b, \infty)$. In this case, the solution is $(-\infty, -3) \cup (3, \infty)$.

Common Mistakes

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When solving absolute value inequalities, there are several common mistakes to avoid. These include:

• Not including the endpoints in the solution.
• Expressing the solution as an interval with the wrong endpoints.
• Not following the correct steps to solve the inequality.

Tips and Tricks

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To solve absolute value inequalities, follow these tips and tricks:

• Use the notation $(-\infty, a) \cup (b, \infty)$ to express the solution as an interval.
• Make sure to include the endpoints in the solution.
• Follow the correct steps to solve the inequality.

Real-World Applications

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Absolute value inequalities have several real-world applications. These include:

• Modeling real-world situations that involve absolute values.
• Solving problems that involve absolute values.
• Expressing solutions as intervals.

Conclusion

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In conclusion, Fuaad's solution to the absolute value inequality is $(-\infty, -12) \cup (7, \infty)$. This solution is expressed as an interval, where the endpoints are $-12$ and $7$. The solution is all values of $x$ that satisfy the inequality $-12 \ \textless \ x \ \textless \ 7$.

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Q: What is an absolute value inequality?

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A: An absolute value inequality is an inequality that involves the absolute value of a variable or expression. It is of the form $|x| \ \textless \ k$, where $k$ is a positive constant.

Q: How do I solve an absolute value inequality?

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A: To solve an absolute value inequality, you need to follow these steps:

1. Write the inequality in the form $|x| \ \textless \ k$.
2. Determine the value of $k$.
3. Express the solution as an interval using the notation $(-\infty, a) \cup (b, \infty)$.

Q: What is the solution to the absolute value inequality $|x| \ \textless \ 3$?

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A: The solution to the absolute value inequality $|x| \ \textless \ 3$ is $(-\infty, -3) \cup (3, \infty)$.

Q: How do I express the solution to an absolute value inequality as an interval?

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A: To express the solution to an absolute value inequality as an interval, you need to use the notation $(-\infty, a) \cup (b, \infty)$, where $a$ and $b$ are the endpoints of the interval.

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

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

• Not including the endpoints in the solution.
• Expressing the solution as an interval with the wrong endpoints.
• Not following the correct steps to solve the inequality.

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

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A: Absolute value inequalities have several real-world applications, including:

• Modeling real-world situations that involve absolute values.
• Solving problems that involve absolute values.
• Expressing solutions as intervals.

Q: How do I use absolute value inequalities in real-world problems?

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A: To use absolute value inequalities in real-world problems, you need to follow these steps:

1. Identify the absolute value expression in the problem.
2. Write the inequality in the form $|x| \ \textless \ k$.
3. Determine the value of $k$.
4. Express the solution as an interval using the notation $(-\infty, a) \cup (b, \infty)$.

Q: What are some examples of real-world problems that involve absolute value inequalities?

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A: Some examples of real-world problems that involve absolute value inequalities include:

• Modeling the distance between two objects.
• Solving problems that involve absolute values in finance.
• Expressing solutions as intervals in physics.

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

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A: To check your solution to an absolute value inequality, you need to follow these steps:

1. Plug in the endpoints of the interval into the original inequality.
2. Check if the inequality is true for the endpoints.
3. If the inequality is true for the endpoints, then the solution is correct.

Q: What are some tips and tricks for solving absolute value inequalities?

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A: Some tips and tricks for solving absolute value inequalities include:

• Use the notation $(-\infty, a) \cup (b, \infty)$ to express the solution as an interval.
• Make sure to include the endpoints in the solution.
• Follow the correct steps to solve the inequality.

Q: How do I practice solving absolute value inequalities?

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A: To practice solving absolute value inequalities, you can try the following:

• Work on practice problems that involve absolute value inequalities.
• Use online resources to practice solving absolute value inequalities.
• Join a study group to practice solving absolute value inequalities with others.