Solve: $3|x-6|+4\ \textgreater \ 12$Please Answer In Interval Notation. If Your Answer Contains A Fraction, Do Not Convert It To A Decimal. Make Sure All Answers Are Simplified.$\square$

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Introduction

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Absolute value inequalities are a type of mathematical problem that involves solving equations and inequalities with absolute values. In this article, we will focus on solving the absolute value inequality $3|x-6|+4 > 12$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding Absolute Value

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Before we dive into solving the inequality, let's take a moment to understand what absolute value means. The absolute value of a number is its distance from zero on the number line. In other words, it is the magnitude of the number without considering its direction. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

Step 1: Subtract 4 from Both Sides

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To solve the inequality $3|x-6|+4 > 12$, we need to isolate the absolute value expression. We can start by subtracting 4 from both sides of the inequality.

3|x-6|+4 > 12
3|x-6| > 8

Step 2: Divide Both Sides by 3

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Next, we need to get rid of the coefficient 3 that is multiplying the absolute value expression. We can do this by dividing both sides of the inequality by 3.

3|x-6| > 8
|x-6| > 8/3

Step 3: Write Two Separate Inequalities

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Now that we have the absolute value expression isolated, we can write two separate inequalities based on the definition of absolute value.

|x-6| > 8/3

This means that either $x-6 > 8/3$ or $x-6 < -8/3$.

Step 4: Solve the First Inequality

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Let's start by solving the first inequality $x-6 > 8/3$.

x-6 > 8/3
x > 6 + 8/3
x > 6 + 2.67
x > 8.67

Step 5: Solve the Second Inequality

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Next, let's solve the second inequality $x-6 < -8/3$.

x-6 < -8/3
x < 6 - 8/3
x < 6 - 2.67
x < 3.33

Step 6: Write the Solution in Interval Notation

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Now that we have solved both inequalities, we can write the solution in interval notation.

x \in (-\infty, 3.33) \cup (8.67, \infty)

This means that the solution is all real numbers less than 3.33 or greater than 8.67.

Conclusion

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Solving absolute value inequalities requires breaking down the problem into manageable steps and using the definition of absolute value to write two separate inequalities. By following these steps, we can solve the inequality $3|x-6|+4 > 12$ and write the solution in interval notation.

Frequently Asked Questions

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• What is the definition of absolute value?
• How do I solve absolute value inequalities?
• What is the solution to the inequality $3|x-6|+4 > 12$?

Final Answer

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The final answer is $\boxed{(-\infty, 3.33) \cup (8.67, \infty)}$.

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Introduction

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Solving absolute value inequalities can be a challenging task, but with the right approach and understanding, it can be made easier. In this article, we will provide a Q&A guide to help you better understand how to solve absolute value inequalities and provide you with a clear explanation of each concept.

Q&A: Absolute Value Inequality Solutions

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Q: What is the definition of absolute value?

A: The absolute value of a number is its distance from zero on the number line. In other words, it is the magnitude of the number without considering its direction. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

Q: How do I solve absolute value inequalities?

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

1. Isolate the absolute value expression.
2. Write two separate inequalities based on the definition of absolute value.
3. Solve each inequality separately.
4. Write the solution in interval notation.

Q: What is the solution to the inequality $3|x-6|+4 > 12$?

A: To solve the inequality $3|x-6|+4 > 12$, we need to follow the steps outlined above. First, we isolate the absolute value expression by subtracting 4 from both sides and then dividing both sides by 3. This gives us $|x-6| > 8/3$. Next, we write two separate inequalities based on the definition of absolute value: $x-6 > 8/3$ and $x-6 < -8/3$. We then solve each inequality separately and write the solution in interval notation.

Q: How do I write the solution in interval notation?

A: To write the solution in interval notation, you need to use the following format:

x \in (a, b)

Where $a$ and $b$ are the endpoints of the interval. For example, if the solution is all real numbers less than 3.33 or greater than 8.67, the solution in interval notation would be:

x \in (-\infty, 3.33) \cup (8.67, \infty)

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

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

• Not isolating the absolute value expression.
• Not writing two separate inequalities based on the definition of absolute value.
• Not solving each inequality separately.
• Not writing the solution in interval notation.

Q: How do I check my solution?

A: To check your solution, you need to plug in a value from each interval into the original inequality and verify that it is true. For example, if the solution is $x \in (-\infty, 3.33) \cup (8.67, \infty)$, you can plug in a value such as $x = 0$ from the first interval and $x = 9$ from the second interval into the original inequality and verify that it is true.

Conclusion

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Solving absolute value inequalities requires a clear understanding of the definition of absolute value and the steps involved in solving the inequality. By following the steps outlined in this article and avoiding common mistakes, you can solve absolute value inequalities with confidence.

Frequently Asked Questions

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• What is the definition of absolute value?
• How do I solve absolute value inequalities?
• What is the solution to the inequality $3|x-6|+4 > 12$?
• How do I write the solution in interval notation?
• What are some common mistakes to avoid when solving absolute value inequalities?
• How do I check my solution?

Final Answer

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The final answer is $\boxed{(-\infty, 3.33) \cup (8.67, \infty)}$.