Solve The Absolute Value Inequality: ${ |3x - 6| \ \textgreater \ 12 }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Solution Set In Interval Notation Is { \square$}$.

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Introduction

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

Understanding Absolute Value Inequalities

Absolute value inequalities involve solving equations with absolute values. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

When solving absolute value inequalities, we need to consider two cases:

  • Case 1: The expression inside the absolute value is positive.
  • Case 2: The expression inside the absolute value is negative.

Solving the Absolute Value Inequality

To solve the absolute value inequality ∣3x−6∣>12|3x - 6| > 12, we need to consider both cases.

Case 1: 3x−6>03x - 6 > 0

In this case, the expression inside the absolute value is positive. We can remove the absolute value sign and solve the inequality.

3x−6>123x - 6 > 12

To solve for x, we need to isolate x on one side of the inequality. We can do this by adding 6 to both sides of the inequality.

3x>183x > 18

Next, we can divide both sides of the inequality by 3 to solve for x.

x>6x > 6

Case 2: 3x−6<03x - 6 < 0

In this case, the expression inside the absolute value is negative. We can remove the absolute value sign and solve the inequality.

−(3x−6)>12-(3x - 6) > 12

To solve for x, we need to isolate x on one side of the inequality. We can do this by multiplying both sides of the inequality by -1.

−3x+6>12-3x + 6 > 12

Next, we can subtract 6 from both sides of the inequality.

−3x>6-3x > 6

Finally, we can divide both sides of the inequality by -3 to solve for x.

x<−2x < -2

Combining the Solutions

We have solved the absolute value inequality ∣3x−6∣>12|3x - 6| > 12 in two cases. In Case 1, we found that x>6x > 6, and in Case 2, we found that x<−2x < -2.

To find the solution set, we need to combine the solutions from both cases. We can do this by using the union symbol, which is represented by the word "or".

The solution set is x>6x > 6 or x<−2x < -2.

Writing the Solution in Interval Notation

Interval notation is a way of writing the solution set using intervals on the number line. We can write the solution set as follows:

(−∞,−2)∪(6,∞)(-\infty, -2) \cup (6, \infty)

This notation indicates that the solution set includes all real numbers less than -2 and all real numbers greater than 6.

Conclusion

Solving absolute value inequalities can be a challenging task, but by breaking down the solution into manageable steps, we can make it more manageable. In this article, we solved the absolute value inequality ∣3x−6∣>12|3x - 6| > 12 in two cases and combined the solutions to find the solution set. We also wrote the solution set in interval notation.

Frequently Asked Questions

Q: What is an absolute value inequality?

A: An absolute value inequality is a type of mathematical problem that involves solving equations with absolute values.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive and when the expression inside the absolute value is negative.

Q: What is the solution set for the absolute value inequality ∣3x−6∣>12|3x - 6| > 12?

A: The solution set for the absolute value inequality ∣3x−6∣>12|3x - 6| > 12 is x>6x > 6 or x<−2x < -2.

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

A: To write the solution set in interval notation, you need to use the union symbol, which is represented by the word "or". The solution set is written as (−∞,−2)∪(6,∞)(-\infty, -2) \cup (6, \infty).

References

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Introduction

In our previous article, we solved the absolute value inequality ∣3x−6∣>12|3x - 6| > 12 and provided a step-by-step guide on how to solve it. However, we know that there are many more questions that you may have about absolute value inequalities. In this article, we will answer some of the most frequently asked questions about absolute value inequalities.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is a type of mathematical problem that involves solving equations with absolute values. It is a statement that involves an absolute value expression and a comparison to a number or expression.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive and when the expression inside the absolute value is negative. You can then solve each case separately and combine the solutions to find the final answer.

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

A: An absolute value equation is a statement that involves an absolute value expression and an equal sign, while an absolute value inequality is a statement that involves an absolute value expression and a comparison to a number or expression.

Q: How do I know which case to use when solving an absolute value inequality?

A: When solving an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive and when the expression inside the absolute value is negative. You can use the following rules to determine which case to use:

  • If the expression inside the absolute value is positive, use the case where the expression is greater than or equal to the number or expression.
  • If the expression inside the absolute value is negative, use the case where the expression is less than or equal to the number or expression.

Q: What is the solution set for the absolute value inequality ∣3x−6∣>12|3x - 6| > 12?

A: The solution set for the absolute value inequality ∣3x−6∣>12|3x - 6| > 12 is x>6x > 6 or x<−2x < -2.

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

A: To write the solution set in interval notation, you need to use the union symbol, which is represented by the word "or". The solution set is written as (−∞,−2)∪(6,∞)(-\infty, -2) \cup (6, \infty).

Q: Can I use a calculator to solve an absolute value inequality?

A: Yes, you can use a calculator to solve an absolute value inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q: How do I check my answer when solving an absolute value inequality?

A: To check your answer when solving an absolute value inequality, you need to plug the solution back into the original inequality and make sure that it is true.

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 considering both cases when solving the inequality.
  • Not using the correct rules to determine which case to use.
  • Not checking the solution to make sure that it is true.

Conclusion

Solving absolute value inequalities can be a challenging task, but by following the steps outlined in this article, you can make it more manageable. Remember to consider both cases when solving the inequality, use the correct rules to determine which case to use, and check the solution to make sure that it is true.

Frequently Asked Questions

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

A: An absolute value inequality is a type of mathematical problem that involves solving equations with absolute values, while a linear inequality is a type of mathematical problem that involves solving equations with linear expressions.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality and make sure that the coefficient of the variable is positive.

Q: Can I use the same steps to solve a linear inequality as I would to solve an absolute value inequality?

A: No, you cannot use the same steps to solve a linear inequality as you would to solve an absolute value inequality. Linear inequalities are solved using different rules and techniques than absolute value inequalities.

References