Manuela Solved The Equation $3 - 2|0.5x + 1.5| = 2$ 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 the equation $3 - 2|0.5x + 1.5| = 2$.

Understanding Absolute Value Equations

An absolute value equation is an 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. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

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

The equation $3 - 2|0.5x + 1.5| = 2$ is an example of an absolute value equation. To solve this equation, we need to isolate the absolute value expression and then solve for the variable $x$.

Step 1: Isolate the Absolute Value Expression

The first step in solving the equation is to isolate the absolute value expression. We can do this by subtracting $3$ from both sides of the equation:

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

Step 2: Remove the Negative Sign

The next step is to remove the negative sign from the equation. We can do this by multiplying both sides of the equation by $-1$:

$$|0.5x + 1.5| = 0.5$$

Step 3: Solve for the Variable

Now that we have isolated the absolute value expression, we can solve for the variable $x$. We can do this by setting up two equations, one for the positive case and one for the negative case:

Positive Case

For the positive case, we have:

$$0.5x + 1.5 = 0.5$$

Subtracting $1.5$ from both sides of the equation gives:

$$0.5x = -1$$

Dividing both sides of the equation by $0.5$ gives:

$$x = -2$$

Negative Case

For the negative case, we have:

$$0.5x + 1.5 = -0.5$$

Subtracting $1.5$ from both sides of the equation gives:

$$0.5x = -2$$

Dividing both sides of the equation by $0.5$ gives:

$$x = -4$$

Conclusion

In this article, we have explored how to solve absolute value equations using the example of the equation $3 - 2|0.5x + 1.5| = 2$. We have seen that to solve an absolute value equation, we need to isolate the absolute value expression and then solve for the variable. We have also seen that absolute value equations can have multiple solutions, depending on the case.

Tips and Tricks

Here are some tips and tricks for solving absolute value equations:

• Isolate the absolute value expression: The first step in solving an absolute value equation is to isolate the absolute value expression.
• Remove the negative sign: The next step is to remove the negative sign from the equation.
• Solve for the variable: Once we have isolated the absolute value expression, we can solve for the variable.
• Check for multiple solutions: Absolute value equations can have multiple solutions, depending on the case.

Common Mistakes

Here are some common mistakes to avoid when solving absolute value equations:

• Not isolating the absolute value expression: Failing to isolate the absolute value expression can make it difficult to solve the equation.
• Not removing the negative sign: Failing to remove the negative sign can lead to incorrect solutions.
• Not checking for multiple solutions: Failing to check for multiple solutions can lead to missing solutions.

Real-World Applications

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

• Physics: Absolute value equations are used to model the motion of objects in physics.
• Engineering: Absolute value equations are used to model the behavior of electrical circuits in engineering.
• Computer Science: Absolute value equations are used to model the behavior of algorithms in computer science.

Conclusion

Introduction

In our previous article, we explored how to solve absolute value equations using the example of the equation $3 - 2|0.5x + 1.5| = 2$. We saw that to solve an absolute value equation, we need to isolate the absolute value expression and then solve for the variable. In this article, we will answer some frequently asked questions about solving absolute value equations.

Q: What is an absolute value equation?

A: An absolute value equation is an 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 and then solve for the variable. You can do this by setting up two equations, one for the positive case and one for the negative case.

Q: What is the positive case?

A: The positive case is when the expression inside the absolute value bars is positive. For example, if we have the equation $|x + 2| = 3$, the positive case is when $x + 2 = 3$.

Q: What is the negative case?

A: The negative case is when the expression inside the absolute value bars is negative. For example, if we have the equation $|x + 2| = 3$, the negative case is when $x + 2 = -3$.

Q: How do I know which case to use?

A: To determine which case to use, you need to look at the expression inside the absolute value bars. If the expression is positive, use the positive case. If the expression is negative, use the negative case.

Q: What if I get two different solutions?

A: If you get two different solutions, it means that the equation has two solutions. You need to check both solutions to make sure they are correct.

Q: What if I get no solutions?

A: If you get no solutions, it means that the equation has no solutions. This can happen if the expression inside the absolute value bars is always positive or always negative.

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. For example, you can use absolute value equations to model the motion of objects in physics or the behavior of electrical circuits in engineering.

Q: Are there any common mistakes to avoid when solving absolute value equations?

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

• Not isolating the absolute value expression
• Not removing the negative sign
• Not checking for multiple solutions

Conclusion

In conclusion, solving absolute value equations can be challenging, but with the right approach, they can be tackled with ease. By isolating the absolute value expression, removing the negative sign, and solving for the variable, we can solve absolute value equations. Additionally, by checking for multiple solutions and avoiding common mistakes, we can ensure that we find all the solutions to an absolute value equation.

Tips and Tricks

Here are some tips and tricks for solving absolute value equations:

• Isolate the absolute value expression: The first step in solving an absolute value equation is to isolate the absolute value expression.
• Remove the negative sign: The next step is to remove the negative sign from the equation.
• Solve for the variable: Once we have isolated the absolute value expression, we can solve for the variable.
• Check for multiple solutions: Absolute value equations can have multiple solutions, depending on the case.

Common Mistakes

Here are some common mistakes to avoid when solving absolute value equations:

• Not isolating the absolute value expression: Failing to isolate the absolute value expression can make it difficult to solve the equation.
• Not removing the negative sign: Failing to remove the negative sign can lead to incorrect solutions.
• Not checking for multiple solutions: Failing to check for multiple solutions can lead to missing solutions.

Real-World Applications

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

• Physics: Absolute value equations are used to model the motion of objects in physics.
• Engineering: Absolute value equations are used to model the behavior of electrical circuits in engineering.
• Computer Science: Absolute value equations are used to model the behavior of algorithms in computer science.