 
Solve the absolute value equation: |3x−1|=13x-1=1

 x=0 or x=−23x=0 or x=-23


x=75 or x=23x=75 or x=23

x=−75 or x=−23x=-75 or x=-23


x=0 or x=23

The equation provided seems a bit unclear. Based on how it's written, I'll interpret the equation as:

\[
|3x - 1| = 1
\]

This is an absolute value equation. To solve it, we'll break it down into two cases:

1. \( 3x - 1 = 1 \)
2. \( 3x - 1 = -1 \)

### Case 1: \( 3x - 1 = 1 \)
Solve for \( x \):
\[
3x - 1 = 1
\]
Add 1 to both sides:
\[
3x = 2
\]
Now divide by 3:
\[
x = \frac{2}{3}
\]

### Case 2: \( 3x - 1 = -1 \)
Solve for \( x \):
\[
3x - 1 = -1
\]
Add 1 to both sides:
\[
3x = 0
\]
Now divide by 3:
\[
x = 0
\]

### Final Solution:
The solutions are:
\[
x = 0 \quad \text{or} \quad x = \frac{2}{3}
\]

From your list of options, the correct one is:

\[
x = 0 \quad \text{or} \quad x = \frac{2}{3}
\]

Would you like further details on how absolute value equations are solved, or have any questions?

Here are some related questions:
1. How do you solve absolute value inequalities?
2. What are the properties of absolute values in equations?
3. How do you solve equations involving multiple absolute values?
4. What is the graphical interpretation of an absolute value function?
5. How do absolute value equations relate to distance?

**Tip**: Always remember that solving absolute value equations leads to two possible cases based on the definition of absolute value!

Solve the absolute value equation: |3x−1|=1

Learn how to solve the absolute value equation |3x−1|=1 using algebraic methods. The detailed solution explores the absolute value theorem and walks through two cases: 3x - 1 = 1 and 3x - 1 = -1. Suitable for students in Grades 8-10.

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