Manuela Solved The Equation 3 − 2 ∣ 0.5 X + 1.5 ∣ = 2 3 - 2|0.5x + 1.5| = 2 3 − 2∣0.5 X + 1.5∣ = 2 For One Solution. Her Work Is Shown Below.$[ \begin{aligned} 3 - 2|0.5x + 1.5| & = 2 \ -2|0.5x + 1.5| & = -1 \ |0.5x + 1.5| & = 0.5 \ 0.5x + 1.5 & = 0.5 \ 0.5x & = -1 \ x & =

Introduction

In mathematics, absolute value equations are a type of equation that involves the absolute value of an expression. These equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore how to solve absolute value equations, using the example of Manuela's work on the equation $3 - 2|0.5x + 1.5| = 2$.

Understanding Absolute Value Equations

Absolute value equations involve the absolute value of an expression, which is denoted by the symbol $| |$. The absolute value of an expression is its distance from zero on the number line, without considering direction. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

The Equation $3 - 2|0.5x + 1.5| = 2$

Manuela's work on the equation $3 - 2|0.5x + 1.5| = 2$ is shown below:

$${ \begin{aligned} 3 - 2|0.5x + 1.5| & = 2 \\ -2|0.5x + 1.5| & = -1 \\ |0.5x + 1.5| & = 0.5 \\ 0.5x + 1.5 & = 0.5 \\ 0.5x & = -1 \\ x & = \end{aligned} }$$

Step 1: Isolate the Absolute Value Expression

The first step in solving an absolute value equation is to isolate the absolute value expression. In this case, we can start by subtracting $3$ from both sides of the equation:

$-2|0.5x + 1.5| = -1$

Step 2: Divide Both Sides by $-2$

Next, we can divide both sides of the equation by $-2$ to isolate the absolute value expression:

$|0.5x + 1.5| = 0.5$

Step 3: 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 $0.5x + 1.5$ is Positive

When $0.5x + 1.5$ is positive, we can remove the absolute value by simply removing the absolute value symbol:

$0.5x + 1.5 = 0.5$

Case 2: When $0.5x + 1.5$ is Negative

When $0.5x + 1.5$ is negative, we can remove the absolute value by multiplying both sides of the equation by $-1$:

$-(0.5x + 1.5) = -0.5$

Step 4: Solve for $x$

Now that we have removed the absolute value, we can solve for $x$ in each case.

Case 1: When $0.5x + 1.5$ is Positive

Subtracting $1.5$ from both sides of the equation, we get:

$0.5x = -1$

Dividing both sides of the equation by $0.5$, we get:

$x = -2$

Case 2: When $0.5x + 1.5$ is Negative

Subtracting $1.5$ from both sides of the equation, we get:

$-0.5x = -1.5$

Dividing both sides of the equation by $-0.5$, we get:

$x = 3$

Conclusion

In this article, we have explored how to solve absolute value equations, using the example of Manuela's work on the equation $3 - 2|0.5x + 1.5| = 2$. We have seen that the first step in solving an absolute value equation is to isolate the absolute value expression, and then remove the absolute value by considering two cases: when the expression inside the absolute value is positive, and when it is negative. We have also seen that the solution to the equation depends on the value of the expression inside the absolute value.

Common Mistakes to Avoid

When solving absolute value equations, there are several common mistakes to avoid. These include:

• Not isolating the absolute value expression
• Not considering both cases when removing the absolute value
• Not checking the validity of the solution

Real-World Applications

Absolute value equations have many real-world applications, including:

• Physics: Absolute value equations are used to model the motion of objects, such as the position of a particle at a given time.
• Engineering: Absolute value equations are used to model the behavior of electrical circuits, such as the voltage across a resistor.
• Economics: Absolute value equations are used to model the behavior of economic systems, such as the price of a commodity.

Practice Problems

Here are some practice problems to help you reinforce your understanding of absolute value equations:

1. Solve the equation $2|3x - 2| = 10$.
2. Solve the equation $|x + 2| = 3$.
3. Solve the equation $|2x - 1| = 4$.

Conclusion

Introduction

In our previous article, we explored how to solve absolute value equations, using the example of Manuela's work on the equation $3 - 2|0.5x + 1.5| = 2$. In this article, we will answer some frequently asked questions about absolute value equations, providing additional insights and examples to help you master this topic.

Q: What is an absolute value equation?

A: An absolute value equation is a type of equation that involves the absolute value of an expression. The absolute value of an expression is its distance from zero on the number line, without considering direction.

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 by considering two cases (when the expression inside the absolute value is positive and when it is negative), and check the validity of the solution.

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

A: When removing the absolute value, you need to consider two cases:

• 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 use?

A: To determine which case to use, you need to evaluate the expression inside the absolute value. If the expression is positive, use Case 1. If the expression is negative, use Case 2.

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

A: An absolute value equation is a type of equation that involves the absolute value of an expression, while a linear equation is a type of equation that involves a linear expression. Absolute value equations can have multiple solutions, while linear equations typically have a single solution.

Q: Can I use algebraic manipulations to solve absolute value equations?

A: Yes, you can use algebraic manipulations to solve absolute value equations. However, you need to be careful when using these manipulations, as they can lead to incorrect solutions.

Q: How do I check the validity of the solution?

A: To check the validity of the solution, you need to plug the solution back into the original equation and verify that it is true.

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 when removing the absolute value
• Not checking the validity of the solution

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

A: Yes, absolute value equations can be used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the behavior of economic systems.

Q: How do I apply absolute value equations to real-world problems?

A: To apply absolute value equations to real-world problems, you need to:

• Identify the variables and constants in the problem
• Write an equation that models the problem
• Solve the equation using absolute value techniques
• Interpret the solution in the context of the problem

Conclusion

In conclusion, absolute value equations are a powerful tool for modeling and solving real-world problems. By understanding how to solve these equations and applying them to real-world problems, you can develop a deeper understanding of mathematics and its applications.

Practice Problems

Here are some practice problems to help you reinforce your understanding of absolute value equations:

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

Additional Resources

For additional resources on absolute value equations, including videos, tutorials, and practice problems, visit the following websites:

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

Conclusion

In conclusion, absolute value equations are a fundamental concept in mathematics that can be used to model and solve real-world problems. By understanding how to solve these equations and applying them to real-world problems, you can develop a deeper understanding of mathematics and its applications.