Solve For \[$ B \$\] In The Equation:$\[ \left|\frac{b}{5}\right| = 1 \\]

$$$$

Introduction

In mathematics, equations involving absolute values can be challenging to solve. The absolute value of a number is its distance from zero on the number line, and it can be positive or negative. In this article, we will focus on solving the equation ${ \left|\frac{b}{5}\right| = 1 }$ for the variable ${ b }$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding Absolute Value Equations

Before we dive into solving the equation, let's understand what absolute value equations are. 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, and it can be positive or negative. For example, the absolute value of -3 is 3, and the absolute value of 3 is also 3.

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 the expression inside the absolute value is negative. In this case, we have the equation ${ \left|\frac{b}{5}\right| = 1 }$. We can start by isolating the expression inside the absolute value.

Isolating the Expression Inside the Absolute Value

To isolate the expression inside the absolute value, we can multiply both sides of the equation by 5. This will give us ${ \left|b\right| = 5 }$.

Solving for ${ b }$

Now that we have isolated the expression inside the absolute value, we can solve for ${ b }$. We have two cases to consider: one where ${ b }$ is positive, and one where ${ b }$ is negative.

Case 1: ${ b }$ is Positive

If ${ b }$ is positive, then the absolute value of ${ b }$ is equal to ${ b }$ itself. We can set up the equation ${ b = 5 }$ and solve for ${ b }$. This gives us ${ b = 5 }$.

Case 2: ${ b }$ is Negative

If ${ b }$ is negative, then the absolute value of ${ b }$ is equal to the negative of ${ b }$ itself. We can set up the equation ${ -b = 5 }$ and solve for ${ b }$. This gives us ${ b = -5 }$.

Conclusion

In this article, we solved the equation ${ \left|\frac{b}{5}\right| = 1 }$ for the variable ${ b }$. We broke down the solution step by step and provided a clear explanation of each step. We considered two cases: one where ${ b }$ is positive, and one where ${ b }$ is negative. We found that the solutions to the equation are ${ b = 5 }$ and ${ b = -5 }$.

Final Answer

The final answer to the equation ${ \left|\frac{b}{5}\right| = 1 }$ is ${ b = 5 }$ or ${ b = -5 }$.

Frequently Asked Questions

• What is an absolute value equation? An absolute value equation is an equation that involves the absolute value of a variable or expression.
• How do you solve an absolute value equation? To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.
• What is the final answer to the equation ${ \left|\frac{b}{5}\right| = 1 }$? The final answer to the equation ${ \left|\frac{b}{5}\right| = 1 }$ is ${ b = 5 }$ or ${ b = -5 }$.

Additional Resources

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

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Linear Algebra" by Jim Hefferon
  

Introduction

In our previous article, we solved the equation ${ \left|\frac{b}{5}\right| = 1 }$ for the variable ${ b }$. We broke down the solution step by step and provided a clear explanation of each step. In this article, we will answer some 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, and it can be positive or negative.

Q: How do you solve an absolute value equation?

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

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

A: An absolute value equation is an equation that involves the absolute value of a variable or expression, while a linear equation is an equation that involves a linear expression. For example, the equation ${ |x| = 3 }$ is an absolute value equation, while the equation ${ x + 2 = 5 }$ is a linear equation.

Q: Can you give an example of an absolute value equation?

A: Yes, here is an example of an absolute value equation: ${ |x - 2| = 4 }$. To solve this equation, you would need to consider two cases: one where ${ x - 2 }$ is positive, and one where ${ x - 2 }$ is negative.

Q: How do you graph an absolute value equation?

A: To graph an absolute value equation, you can use the following steps:

1. Find the vertex of the graph, which is the point where the graph is at its minimum or maximum value.
2. Find the x-intercepts of the graph, which are the points where the graph crosses the x-axis.
3. Plot the graph using the vertex and x-intercepts.

Q: Can you give an example of a real-world application of absolute value equations?

A: Yes, here is an example of a real-world application of absolute value equations: Suppose you are a manager at a company and you need to determine the maximum and minimum values of a certain quantity. You can use an absolute value equation to model the situation and find the maximum and minimum values.

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 cases (positive and negative) when solving the equation.
• Not isolating the expression inside the absolute value.
• Not checking the solutions to make sure they are valid.

Conclusion

In this article, we answered some frequently asked questions about absolute value equations. We covered topics such as what an absolute value equation is, how to solve an absolute value equation, and how to graph an absolute value equation. We also provided some examples of real-world applications of absolute value equations and some common mistakes to avoid when solving them.

Final Answer

The final answer to the question "What is an absolute value equation?" is an equation that involves the absolute value of a variable or expression.

Frequently Asked Questions

• What is an absolute value equation? An absolute value equation is an equation that involves the absolute value of a variable or expression.
• How do you solve an absolute value equation? To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.
• What is the difference between an absolute value equation and a linear equation? An absolute value equation is an equation that involves the absolute value of a variable or expression, while a linear equation is an equation that involves a linear expression.

Additional Resources

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

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Linear Algebra" by Jim Hefferon