Solve The Equation:\[$-3 = -|z-5| + 4\$\]The Solution Set Is \[$\square\$\].

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Introduction

In this article, we will delve into solving the equation $-3 = -|z-5| + 4$. This equation involves absolute value, which can sometimes be challenging to solve. However, with a step-by-step approach, we can break down the problem and find the solution set.

Understanding Absolute Value Equations

Before we dive into solving the equation, let's briefly discuss what absolute value equations are and how to approach them. Absolute value equations are equations that involve the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction.

For example, if we have the equation $|x| = 3$, it means that the distance of $x$ from zero is 3. This can be represented as two separate equations: $x = 3$ and $x = -3$.

Solving the Equation

Now that we have a basic understanding of absolute value equations, let's focus on solving the equation $-3 = -|z-5| + 4$. To start, we need to isolate the absolute value expression.

Step 1: Isolate the Absolute Value Expression

We can start by subtracting 4 from both sides of the equation:

$$-3 - 4 = -|z-5| + 4 - 4$$

This simplifies to:

$$-7 = -|z-5|$$

Step 2: Multiply Both Sides by -1

To get rid of the negative sign in front of the absolute value expression, we can multiply both sides of the equation by -1:

$$-(-7) = -(-|z-5|)$$

This simplifies to:

$$7 = |z-5|$$

Step 3: Solve for z

Now that we have the absolute value expression isolated, we can solve for $z$. Since the absolute value of $z-5$ is 7, we can set up two separate equations:

$$z-5 = 7$$

and

$$z-5 = -7$$

Step 4: Solve the First Equation

Solving the first equation, we get:

$$z-5 = 7$$

Adding 5 to both sides, we get:

$$z = 12$$

Step 5: Solve the Second Equation

Solving the second equation, we get:

$$z-5 = -7$$

Adding 5 to both sides, we get:

$$z = -2$$

Conclusion

In this article, we solved the equation $-3 = -|z-5| + 4$ and found the solution set to be $\boxed{\{-2, 12\}}$. We broke down the problem into smaller steps, isolating the absolute value expression and then solving for $z$. With a step-by-step approach, we can tackle even the most challenging absolute value equations.

Final Answer

The final answer is $\boxed{\{-2, 12\}}$.

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Introduction

In our previous article, we solved the equation $-3 = -|z-5| + 4$ and found the solution set to be $\boxed{\{-2, 12\}}$. However, we understand that some readers may still have questions or need further clarification on the steps involved in solving the equation. In this article, we will address some of the most frequently asked questions related to solving absolute value equations.

Q&A

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

A: An absolute value equation is an equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. Regular equations, on the other hand, do not involve absolute value.

Q: How do I know which direction to go when solving an absolute value equation?

A: When solving an absolute value equation, you need to consider two separate equations: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. This is because the absolute value of a number is its distance from zero, regardless of direction.

Q: Can I use the same steps to solve all absolute value equations?

A: While the steps involved in solving absolute value equations are similar, the specific steps may vary depending on the equation. For example, if the equation involves a negative sign in front of the absolute value expression, you may need to multiply both sides by -1 to get rid of the negative sign.

Q: What if I get stuck on a step or don't understand the solution?

A: Don't worry! Solving absolute value equations can be challenging, but with practice and patience, you can master it. If you get stuck on a step or don't understand the solution, try breaking down the problem into smaller steps or seeking help from a teacher or tutor.

Q: Can I use absolute value equations in real-life situations?

A: Yes! Absolute value equations have many real-life applications, such as modeling distance, speed, and time. For example, if you're driving a car and you want to know how far you are from a certain location, you can use an absolute value equation to calculate the distance.

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

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

• Not considering both directions of the absolute value expression
• Not isolating the absolute value expression correctly
• Not multiplying both sides by -1 when necessary
• Not checking the solution for extraneous solutions

Conclusion

In this article, we addressed some of the most frequently asked questions related to solving absolute value equations. We hope that this Q&A article has provided you with a better understanding of how to solve absolute value equations and has helped you to overcome any challenges you may have faced.

Final Tips

• Practice, practice, practice! Solving absolute value equations takes practice, so make sure to work on plenty of problems to build your skills.
• Break down the problem into smaller steps. This will help you to understand the solution and avoid mistakes.
• Don't be afraid to ask for help. If you're stuck on a step or don't understand the solution, don't hesitate to ask a teacher or tutor for help.

Final Answer

The final answer is $\boxed{\{-2, 12\}}$.