Solve: 9 + 6 ∣ 3 X − 3 ∣ \textless 39 9 + 6|3x - 3| \ \textless \ 39 9 + 6∣3 X − 3∣ \textless 39 □ \square □

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Introduction

In this article, we will delve into the world of mathematical inequalities and solve the given expression. The expression involves absolute value, which can sometimes be challenging to work with. However, with the right approach and techniques, we can simplify the expression and find the solution.

Understanding the Expression

The given expression is $9 + 6|3x - 3| \ \textless \ 39$. This expression involves an absolute value term, which is $|3x - 3|$. The absolute value of a number is its distance from zero on the number line, without considering whether it's positive or negative. In this case, the absolute value term is being multiplied by 6 and then added to 9.

Isolating the Absolute Value Term

To solve this expression, we need to isolate the absolute value term. We can do this by subtracting 9 from both sides of the inequality. This gives us:

$6|3x - 3| \ \textless \ 30$

Dividing by 6

Next, we can divide both sides of the inequality by 6 to get:

$|3x - 3| \ \textless \ 5$

Understanding the Absolute Value Inequality

Now that we have isolated the absolute value term, we need to understand what this inequality means. The absolute value of a number is less than 5 if and only if the number is between -5 and 5, not including -5 and 5. In other words, we have:

$-5 \ \textless \ 3x - 3 \ \textless \ 5$

Solving the Inequality

To solve this inequality, we can add 3 to all three parts of the inequality. This gives us:

$-2 \ \textless \ 3x \ \textless \ 8$

Dividing by 3

Finally, we can divide all three parts of the inequality by 3 to get:

$-\frac{2}{3} \ \textless \ x \ \textless \ \frac{8}{3}$

Conclusion

In this article, we solved the given expression $9 + 6|3x - 3| \ \textless \ 39$. We started by isolating the absolute value term, then divided by 6 to get $|3x - 3| \ \textless \ 5$. We then understood what this inequality means and solved it to get $-\frac{2}{3} \ \textless \ x \ \textless \ \frac{8}{3}$.

Tips and Tricks

• When working with absolute value inequalities, it's essential to remember that the absolute value of a number is its distance from zero on the number line.
• To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and one where it's negative.
• When dividing by a number, make sure to divide all three parts of the inequality by that number.

Real-World Applications

• Absolute value inequalities are used in many real-world applications, such as physics and engineering.
• They can be used to model real-world situations, such as the distance between two objects or the speed of an object.
• Absolute value inequalities can also be used to solve problems involving finance and economics.

Common Mistakes

• One common mistake when working with absolute value inequalities is to forget to consider both cases: one where the expression inside the absolute value is positive, and one where it's negative.
• Another common mistake is to divide by a number without checking if it's zero.
• It's also essential to remember to divide all three parts of the inequality by the same number.

Final Thoughts

In conclusion, solving absolute value inequalities can be challenging, but with the right approach and techniques, we can simplify the expression and find the solution. Remember to isolate the absolute value term, understand what the inequality means, and solve it by considering both cases. With practice and patience, you'll become proficient in solving absolute value inequalities and be able to apply them to real-world problems.

Introduction

In our previous article, we solved the expression $9 + 6|3x - 3| \ \textless \ 39$. We also discussed the basics of absolute value inequalities and how to solve them. In this article, we will answer some frequently asked questions about absolute value inequalities.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves the absolute value of a variable or expression. It is a mathematical statement that describes a relationship between a variable and a constant, where the variable is enclosed in absolute value symbols.

Q: How do I solve an absolute value inequality?

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

1. Isolate the absolute value term.
2. Understand what the inequality means.
3. Solve the inequality by considering both cases: one where the expression inside the absolute value is positive, and one where it's negative.

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

A: The main difference between an absolute value inequality and a regular inequality is the presence of absolute value symbols. In a regular inequality, the variable is not enclosed in absolute value symbols, whereas in an absolute value inequality, the variable is enclosed in absolute value symbols.

Q: Can I use the same methods to solve absolute value inequalities as I would for regular inequalities?

A: No, you cannot use the same methods to solve absolute value inequalities as you would for regular inequalities. Absolute value inequalities require a different approach, as you need to consider both cases: one where the expression inside the absolute value is positive, and one where it's negative.

Q: How do I know which case to consider first?

A: When solving an absolute value inequality, you need to consider both cases: one where the expression inside the absolute value is positive, and one where it's negative. You can start by considering the case where the expression inside the absolute value is positive, and then consider the case where it's negative.

Q: Can I use algebraic manipulations to solve absolute value inequalities?

A: Yes, you can use algebraic manipulations to solve absolute value inequalities. However, you need to be careful when using algebraic manipulations, as they can sometimes lead to incorrect solutions.

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

A: To check your solutions to an absolute value inequality, you need to plug the solutions back into the original inequality and verify that they satisfy the inequality.

Q: Can I use technology to solve absolute value inequalities?

A: Yes, you can use technology to solve absolute value inequalities. Many graphing calculators and computer algebra systems can solve absolute value inequalities and provide the solutions.

Tips and Tricks

• When solving absolute value inequalities, it's essential to consider both cases: one where the expression inside the absolute value is positive, and one where it's negative.
• Use algebraic manipulations carefully, as they can sometimes lead to incorrect solutions.
• Check your solutions to an absolute value inequality by plugging them back into the original inequality.
• Use technology to solve absolute value inequalities, especially when the inequality is complex.

Real-World Applications

• Absolute value inequalities are used in many real-world applications, such as physics and engineering.
• They can be used to model real-world situations, such as the distance between two objects or the speed of an object.
• Absolute value inequalities can also be used to solve problems involving finance and economics.

Common Mistakes

• One common mistake when solving absolute value inequalities is to forget to consider both cases: one where the expression inside the absolute value is positive, and one where it's negative.
• Another common mistake is to use algebraic manipulations without checking the solutions.
• It's also essential to remember to check the solutions to an absolute value inequality by plugging them back into the original inequality.

Final Thoughts

In conclusion, solving absolute value inequalities can be challenging, but with the right approach and techniques, we can simplify the expression and find the solution. Remember to consider both cases, use algebraic manipulations carefully, and check the solutions by plugging them back into the original inequality. With practice and patience, you'll become proficient in solving absolute value inequalities and be able to apply them to real-world problems.