https://file.aiquickdraw.com/mathbot/file/1726450229844s23kk59l.jpg

The equation given is \(|2x - 1| = |4x + 1|\), and the graph visualizes the absolute value functions for each side of the equation.

### Steps to solve the equation using the graph:

1. **Intersection Points**: The solution to \(|2x - 1| = |4x + 1|\) occurs where the two graphs intersect.
   - The red line corresponds to the graph of \( y = |2x - 1| \).
   - The blue line corresponds to \( y = |4x + 1| \).

2. **Identify Intersection Points**: 
   - The graph shows two points of intersection at \(x = 0\) and \(x = 1\). These points solve the equation because the values of \( |2x - 1| \) and \( |4x + 1| \) are equal at these \(x\)-coordinates.

Thus, the solutions are:
\[
x = 0 \quad \text{and} \quad x = 1
\]

Would you like a more detailed breakdown of this process, or do you have any specific questions?

Here are 5 follow-up questions for deeper exploration:
1. How would you solve this equation algebraically, without using the graph?
2. Why do absolute value equations often have two possible solutions?
3. How can you check the solutions \(x = 0\) and \(x = 1\) by substituting them into the original equation?
4. What happens if you shift or scale the absolute value functions on the graph?
5. How do absolute value graphs change when constants are added inside or outside the absolute value?

**Tip**: When solving absolute value equations, always consider the two cases where the expressions inside the absolute values are equal and where their negatives are equal!

Use the graph to solve: |2x - 1| = |4x + 1|.

Learn how to solve the absolute value equation |2x - 1| = |4x + 1| using graphical methods. This problem demonstrates the use of graph intersections to find solutions for absolute value equations. Suitable for students in Grades 9-12.

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