Solve For $x$: $\[ 2|3x + 1| - 4 = 6 \\]A. $x = -\frac{2}{3}$ Or $x = 0$B. $x = 2$ Or $x = 0$C. $x = \frac{4}{3}$ Or $x = -2$D. $x = \frac{4}{3}$ Or $x =

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Introduction

In this article, we will be solving a linear equation that involves absolute value. The equation is given as $2|3x + 1| - 4 = 6$. Our goal is to find the value of $x$ that satisfies this equation. We will use algebraic techniques to solve for $x$ and determine the correct solution.

Understanding Absolute Value Equations

Before we dive into solving the equation, let's review what absolute value equations are. An absolute value equation is an equation that involves the absolute value of a quantity. The absolute value of a quantity is its distance from zero on the number line, without considering direction. In other words, the absolute value of a number is always non-negative.

Step 1: Isolate the Absolute Value Expression

To solve the equation, we need to isolate the absolute value expression. We can do this by adding 4 to both sides of the equation, which gives us:

${ 2|3x + 1| = 10 }$

Step 2: Divide Both Sides by 2

Next, we can divide both sides of the equation by 2 to get:

${ |3x + 1| = 5 }$

Step 3: Set Up Two Equations

Since the absolute value of a quantity can be positive or negative, we need to set up two equations to account for both possibilities. We can set up the following two equations:

${ 3x + 1 = 5 }$ ${ 3x + 1 = -5 }$

Step 4: Solve the First Equation

Let's solve the first equation:

${ 3x + 1 = 5 }$

Subtracting 1 from both sides gives us:

${ 3x = 4 }$

Dividing both sides by 3 gives us:

${ x = \frac{4}{3} }$

Step 5: Solve the Second Equation

Now, let's solve the second equation:

${ 3x + 1 = -5 }$

Subtracting 1 from both sides gives us:

${ 3x = -6 }$

Dividing both sides by 3 gives us:

${ x = -2 }$

Conclusion

We have found two possible values for $x$: $x = \frac{4}{3}$ and $x = -2$. Therefore, the correct answer is:

C. $x = \frac{4}{3}$ or $x = -2$

This solution is based on the algebraic techniques used to solve the absolute value equation. We have isolated the absolute value expression, set up two equations, and solved for $x$ in each equation. The final answer is a combination of the two possible values for $x$.

Discussion

The solution to this problem involves understanding absolute value equations and using algebraic techniques to solve for $x$. The key concept is to isolate the absolute value expression and set up two equations to account for both possibilities. By solving each equation, we can find the possible values for $x$.

Final Answer

The final answer is C. $x = \frac{4}{3}$ or $x = -2$.

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Introduction

In our previous article, we solved the linear equation $2|3x + 1| - 4 = 6$ and found that the possible values for $x$ are $x = \frac{4}{3}$ and $x = -2$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving 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 quantity. The absolute value of a quantity is its distance from zero on the number line, without considering direction. In other words, the absolute value of a number is always non-negative.

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

A: To isolate the absolute value expression, you need to get the absolute value term by itself on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation.

Q: What is the difference between the two equations when solving an absolute value equation?

A: When solving an absolute value equation, you need to set up two equations to account for both possibilities. The first equation is obtained by setting the expression inside the absolute value equal to the positive value, and the second equation is obtained by setting the expression inside the absolute value equal to the negative value.

Q: How do I solve the two equations when solving an absolute value equation?

A: To solve the two equations, you need to follow the same steps as solving a linear equation. You can add or subtract the same value to both sides of the equation, and then divide both sides by the coefficient of the variable.

Q: What if I get two different values for $x$ when solving an absolute value equation?

A: If you get two different values for $x$ when solving an absolute value equation, then both values are possible solutions. You need to check both values in the original equation to make sure they are true.

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

A: Yes, you can use a calculator to solve an absolute value equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.

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 setting up two equations to account for both possibilities
• Not following the same steps as solving a linear equation
• Not checking the solution in the original equation

Conclusion

Solving absolute value equations can be challenging, but with practice and patience, you can master this skill. Remember to isolate the absolute value expression, set up two equations, and solve for $x$ in each equation. By following these steps and avoiding common mistakes, you can find the possible values for $x$ and solve the equation.

Final Answer

The final answer is C. $x = \frac{4}{3}$ or $x = -2$.