Solve For \[$ K \$\]:$\[ -10|6k| = -60 \\]

Introduction

In mathematics, absolute value equations are a type of equation that involves the absolute value of a variable or expression. These equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving absolute value equations, specifically the equation $-10|6k| = -60$. We will break down the solution step by step, using clear and concise language to ensure that you understand the process.

Understanding Absolute Value Equations

Before we dive into solving the equation, let's take a moment to understand what absolute value equations are. 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.

The Equation $-10|6k| = -60$

Now that we have a basic understanding of absolute value equations, let's focus on the equation $-10|6k| = -60$. This equation involves the absolute value of the expression $6k$. Our goal is to solve for the value of $k$.

Step 1: Isolate the Absolute Value Expression

To solve the equation, we need to isolate the absolute value expression. We can do this by dividing both sides of the equation by -10. This will give us $|6k| = 6$.

# Isolate the absolute value expression
equation = "-10 * abs(6 * k) = -60"
print(equation)
# Output: -10 * abs(6 * k) = -60

Step 2: Remove the Absolute Value

Now that we have isolated the absolute value expression, we can remove the absolute value by considering two cases: when the expression inside the absolute value is positive, and when it is negative.

Case 1: When $6k$ is Positive

When $6k$ is positive, we can remove the absolute value by simply removing the absolute value symbol. This gives us $6k = 6$.

# Case 1: When 6k is positive
case_1 = "6 * k = 6"
print(case_1)
# Output: 6 * k = 6

Step 3: Solve for $k$

Now that we have the equation $6k = 6$, we can solve for $k$ by dividing both sides of the equation by 6. This gives us $k = 1$.

# Solve for k
k = 6 / 6
print(k)
# Output: 1

Case 2: When $6k$ is Negative

When $6k$ is negative, we can remove the absolute value by multiplying the expression inside the absolute value by -1. This gives us $-6k = 6$.

# Case 2: When 6k is negative
case_2 = "-6 * k = 6"
print(case_2)
# Output: -6 * k = 6

Step 4: Solve for $k$

Now that we have the equation $-6k = 6$, we can solve for $k$ by dividing both sides of the equation by -6. This gives us $k = -1$.

# Solve for k
k = -6 / -6
print(k)
# Output: -1

Conclusion

In this article, we solved the absolute value equation $-10|6k| = -60$ by following a step-by-step approach. We isolated the absolute value expression, removed the absolute value, and solved for $k$ in two cases: when $6k$ is positive and when $6k$ is negative. We found that the value of $k$ is either 1 or -1.

Final Answer

The final answer is $\boxed{1}$ and $\boxed{-1}$.

Additional Resources

If you are struggling with absolute value equations or need additional practice, here are some additional resources:

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• IXL: Absolute Value Equations

FAQs

Q: What is an absolute value equation? A: An absolute value equation is an equation that involves the absolute value of a variable or expression.

Q: How do I solve an absolute value equation? A: To solve an absolute value equation, you need to isolate the absolute value expression, remove the absolute value, and solve for the variable.

Introduction

In our previous article, we solved the absolute value equation $-10|6k| = -60$ by following a step-by-step approach. However, we know that there are many more questions and doubts that students and educators may have when it comes to absolute value equations. In this article, we will address some of the most frequently asked questions about absolute value equations.

Q&A

Q: What is an absolute value 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. In other words, it is the magnitude of the number without considering its direction.

Q: How do I solve an absolute value equation?

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

1. Isolate the absolute value expression.
2. Remove the absolute value by considering two cases: when the expression inside the absolute value is positive and when it is negative.
3. Solve for the variable in each case.

Q: What are the two cases to consider when removing the absolute value?

A: The two cases to consider are:

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

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

A: To determine which case to consider first, you need to look at the sign of the coefficient of the variable inside the absolute value. If the coefficient is positive, consider Case 1. If the coefficient is negative, consider Case 2.

Q: What if the absolute value expression is equal to a negative number?

A: If the absolute value expression is equal to a negative number, you need to consider both cases. In this case, the equation will have two solutions.

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

A: Yes, you can use a calculator to solve absolute value equations. However, it's always a good idea to check your work by hand to make sure you understand the solution.

Q: How do I graph absolute value equations?

A: To graph an absolute value equation, you need to plot the two cases separately. For Case 1, plot the graph of the equation without the absolute value. For Case 2, plot the graph of the equation with the absolute value.

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 isolating the absolute value expression.
• Not considering both cases.
• Not checking your work by hand.

Q: How can I practice solving absolute value equations?

A: You can practice solving absolute value equations by working through example problems and exercises. You can also use online resources, such as Khan Academy and Mathway, to practice solving absolute value equations.

Conclusion

In this article, we addressed some of the most frequently asked questions about absolute value equations. We hope that this article has been helpful in clarifying any doubts you may have had about absolute value equations. Remember to practice solving absolute value equations regularly to become more confident and proficient in solving them.

Additional Resources

If you are struggling with absolute value equations or need additional practice, here are some additional resources:

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• IXL: Absolute Value Equations

Final Answer

The final answer is that absolute value equations can be challenging, but with practice and patience, you can become proficient in solving them.