Solve The Equation $|1-3t|+2=9$.Select The Correct Choice Below And Fill In Any Answer Boxes Within Your Choice.A. The Solution Set Is $\square$. (Use A Comma To Separate Answers As Needed.)B. The Solution Set Is All Real Numbers.C.

Introduction

Absolute value equations are a type of algebraic equation that involves the absolute value of a variable or expression. These equations can be challenging to solve, but with a clear understanding of the concept and a step-by-step approach, you can master solving absolute value equations. In this article, we will focus on solving the equation $|1-3t|+2=9$.

Understanding Absolute Value

Before we dive into solving the equation, let's review the concept of absolute value. 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. This means that the absolute value of a number is always non-negative.

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.

Case 1: $1-3t \geq 0$

When $1-3t \geq 0$, the absolute value equation becomes:

$$1-3t+2=9$$

Simplifying the equation, we get:

$$3-3t=9$$

Subtracting 3 from both sides, we get:

$$-3t=6$$

Dividing both sides by -3, we get:

$$t=-2$$

Case 2: $1-3t < 0$

When $1-3t < 0$, the absolute value equation becomes:

$$-(1-3t)+2=9$$

Simplifying the equation, we get:

$$-1+3t+2=9$$

Combining like terms, we get:

$$3t+1=9$$

Subtracting 1 from both sides, we get:

$$3t=8$$

Dividing both sides by 3, we get:

$$t=\frac{8}{3}$$

Conclusion

In conclusion, the solution set for the equation $|1-3t|+2=9$ is $t=-2$ and $t=\frac{8}{3}$. These two values satisfy the equation and are the only solutions.

Answer

The correct answer is:

A. The solution set is $-2, \frac{8}{3}$.

Discussion

This problem requires a clear understanding of absolute value and the ability to solve equations involving absolute value. The key to solving this problem is to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. By following these steps, you can master solving absolute value equations and tackle more complex problems with confidence.

Additional Tips

• When solving absolute value equations, always consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.
• Use the properties of absolute value to simplify the equation and isolate the variable.
• Check your solutions by plugging them back into the original equation to ensure that they are true.

Introduction

In our previous article, we explored the concept of absolute value equations and provided a step-by-step guide on how to solve them. However, we know that practice makes perfect, and sometimes, it's helpful to have a Q&A guide to clarify any doubts or questions you may have. In this article, we will address some common questions and provide additional tips on solving absolute value equations.

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

A: An absolute value equation involves the absolute value of a variable or expression, whereas a regular equation does not. Absolute value equations can be challenging to solve because they involve two cases: one where the expression inside the absolute value is positive, and one where it is negative.

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

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, use the first case. If the expression is negative, use the second case.

Q: What if the expression inside the absolute value is zero?

A: If the expression inside the absolute value is zero, then the absolute value equation becomes a regular equation. In this case, you can solve the equation using the usual methods.

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

A: While a calculator can be helpful in solving absolute value equations, it's not always the best approach. Calculators can be prone to errors, and it's often better to solve the equation by hand to ensure accuracy.

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

A: To check your solutions, plug them back into the original equation to ensure that they are true. If the solution satisfies the equation, then it is a valid solution.

Q: What if I get multiple solutions to an absolute value equation?

A: If you get multiple solutions to an absolute value equation, then all of them are valid solutions. However, be sure to check each solution to ensure that it satisfies the equation.

Q: Can I use absolute value equations to solve real-world problems?

A: Yes, absolute value equations can be used to solve real-world problems. For example, in physics, absolute value equations can be used to model the motion of objects. In finance, absolute value equations can be used to model the value of investments.

Additional Tips

• When solving absolute value equations, always consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.
• Use the properties of absolute value to simplify the equation and isolate the variable.
• Check your solutions by plugging them back into the original equation to ensure that they are true.
• Practice solving absolute value equations to become proficient in solving these types of problems.

Common Mistakes to Avoid

• Failing to consider both cases when solving an absolute value equation.
• Not checking solutions to ensure that they satisfy the equation.
• Using a calculator to solve absolute value equations without double-checking the results.

By following these tips and practicing solving absolute value equations, you can become proficient in solving these types of problems and tackle more complex math challenges with confidence.