Solve For $u$. − 4 ∣ 2 U + 8 ∣ = − 64 -4|2u + 8| = -64 − 4∣2 U + 8∣ = − 64 If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, State No Solution. U = U = U =

Introduction

Absolute value equations are a fundamental concept in algebra, and solving them requires a clear understanding of the properties of absolute value. In this article, we will focus on solving absolute value equations of the form $|ax + b| = c$, where $a$, $b$, and $c$ are constants. We will use the given equation $-4|2u + 8| = -64$ as an example to demonstrate the steps involved in solving absolute value equations.

Understanding Absolute Value

Before we dive into solving the equation, let's briefly review the concept of absolute value. The absolute value of a number $x$, denoted by $|x|$, is the distance of $x$ from zero on the number line. In other words, it is the magnitude of $x$ without considering its direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $5$ is also $5$.

Step 1: Isolate the Absolute Value Expression

The first step in solving the equation is to isolate the absolute value expression. In this case, we have $-4|2u + 8| = -64$. To isolate the absolute value expression, we need to divide both sides of the equation by $-4$. This gives us $|2u + 8| = 16$.

Step 2: Write Two Separate Equations

Since the absolute value of an expression can be positive or negative, we need to write two separate equations to account for both possibilities. The first equation is obtained by removing the absolute value sign and changing the sign of the expression inside the absolute value. This gives us $2u + 8 = 16$. The second equation is obtained by removing the absolute value sign and keeping the sign of the expression inside the absolute value. This gives us $2u + 8 = -16$.

Step 3: Solve the First Equation

Now that we have the first equation, we can solve for $u$. Subtracting $8$ from both sides of the equation gives us $2u = 8$. Dividing both sides of the equation by $2$ gives us $u = 4$.

Step 4: Solve the Second Equation

Now that we have the second equation, we can solve for $u$. Subtracting $8$ from both sides of the equation gives us $2u = -24$. Dividing both sides of the equation by $2$ gives us $u = -12$.

Conclusion

In this article, we have demonstrated the steps involved in solving absolute value equations. We used the given equation $-4|2u + 8| = -64$ as an example to illustrate the process. We isolated the absolute value expression, wrote two separate equations, and solved each equation to find the values of $u$. The final answer is $u = 4, -12$.

Additional Tips and Tricks

• When solving absolute value equations, it's essential to remember that the absolute value of an expression can be positive or negative.
• To isolate the absolute value expression, divide both sides of the equation by the coefficient of the absolute value expression.
• When writing two separate equations, remove the absolute value sign and change the sign of the expression inside the absolute value in the first equation, and remove the absolute value sign and keep the sign of the expression inside the absolute value in the second equation.
• When solving each equation, follow the usual steps of algebra, such as adding or subtracting the same value to both sides of the equation and dividing both sides of the equation by the same non-zero value.

Common Mistakes to Avoid

• Failing to isolate the absolute value expression before writing two separate equations.
• Not changing the sign of the expression inside the absolute value in the first equation.
• Not keeping the sign of the expression inside the absolute value in the second equation.
• Not following the usual steps of algebra when solving each equation.

Real-World Applications

Absolute value equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the absolute value of a velocity or acceleration can represent the magnitude of the motion, while in engineering, the absolute value of a signal can represent the amplitude of the signal. In economics, the absolute value of a profit or loss can represent the magnitude of the financial outcome.

Conclusion

Introduction

In our previous article, we discussed the steps involved in solving absolute value equations. In this article, we will provide a Q&A guide to help you better understand the concepts and apply them to real-world problems.

Q: What is an absolute value equation?

A: An absolute value equation is an equation that contains an absolute value expression. The absolute value of an expression is the distance of the expression from zero on the number line.

Q: How do I isolate the absolute value expression in an equation?

A: To isolate the absolute value expression, you need to divide both sides of the equation by the coefficient of the absolute value expression.

Q: What are the two separate equations that I need to write when solving an absolute value equation?

A: The two separate equations are obtained by removing the absolute value sign and changing the sign of the expression inside the absolute value in the first equation, and removing the absolute value sign and keeping the sign of the expression inside the absolute value in the second equation.

Q: How do I solve the first equation?

A: To solve the first equation, you need to follow the usual steps of algebra, such as adding or subtracting the same value to both sides of the equation and dividing both sides of the equation by the same non-zero value.

Q: How do I solve the second equation?

A: To solve the second equation, you need to follow the usual steps of algebra, such as adding or subtracting the same value to both sides of the equation and dividing both sides of the equation by the same non-zero value.

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

A: Some common mistakes to avoid include failing to isolate the absolute value expression before writing two separate equations, not changing the sign of the expression inside the absolute value in the first equation, not keeping the sign of the expression inside the absolute value in the second equation, and not following the usual steps of algebra when solving each equation.

Q: What are some real-world applications of absolute value equations?

A: Absolute value equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the absolute value of a velocity or acceleration can represent the magnitude of the motion, while in engineering, the absolute value of a signal can represent the amplitude of the signal. In economics, the absolute value of a profit or loss can represent the magnitude of the financial outcome.

Q: How do I determine if an absolute value equation has one solution, two solutions, or no solution?

A: To determine if an absolute value equation has one solution, two solutions, or no solution, you need to examine the two separate equations that you wrote. If both equations have the same solution, then the absolute value equation has one solution. If the two equations have different solutions, then the absolute value equation has two solutions. If one of the equations has no solution, then the absolute value equation has no solution.

Q: What is the final answer to the equation $-4|2u + 8| = -64$?

A: The final answer to the equation $-4|2u + 8| = -64$ is $u = 4, -12$.

Conclusion

In conclusion, solving absolute value equations requires a clear understanding of the properties of absolute value and a step-by-step approach. By following the steps outlined in this article and avoiding common mistakes, you can solve absolute value equations with confidence and apply the concepts to real-world problems.

Additional Resources

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• Wolfram Alpha: Absolute Value Equations

Practice Problems

1. Solve the equation $|3x - 2| = 5$.
2. Solve the equation $|2x + 1| = 3$.
3. Solve the equation $|x - 4| = 2$.

Answer Key

1. $x = 3, 1$
2. $x = -2, 2$
3. $x = 6, 2$