https://file.aiquickdraw.com/mathbot/file/17286797524207xubkp0b.jpg

The equation presented in the image is:

\[
5.7x = |2x + 1|
\]

### Solution without a graphing calculator:

This is an equation involving an absolute value, so we need to consider two cases based on the definition of the absolute value:

1. **Case 1**: When \( 2x + 1 \geq 0 \) (the expression inside the absolute value is non-negative):
   \[
   5.7x = 2x + 1
   \]
   Solving this:
   \[
   5.7x - 2x = 1 \implies 3.7x = 1 \implies x = \frac{1}{3.7} \approx 0.27
   \]

2. **Case 2**: When \( 2x + 1 < 0 \) (the expression inside the absolute value is negative, so we take the negative of the expression):
   \[
   5.7x = -(2x + 1) \implies 5.7x = -2x - 1
   \]
   Solving this:
   \[
   5.7x + 2x = -1 \implies 7.7x = -1 \implies x = \frac{-1}{7.7} \approx -0.13
   \]

### Check the validity of solutions:
- For **Case 1**: \( x = 0.27 \), we check \( 2(0.27) + 1 = 1.54 \geq 0 \), which is valid.
- For **Case 2**: \( x = -0.13 \), we check \( 2(-0.13) + 1 = 0.74 \), which should have been negative, but isn't. Thus, **Case 2 is invalid**.

### Final Answer:
\[
x \approx 0.27
\]

Do you have any questions or want more details?

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Here are 5 related questions to further explore the concept:

1. How do you solve absolute value equations algebraically?
2. How do graphing calculators handle absolute value functions visually?
3. What is the significance of breaking absolute value equations into different cases?
4. Can you graph \( y = 5.7x \) and \( y = |2x + 1| \) to verify the solution visually?
5. How do real-world scenarios involve absolute value equations?

**Tip**: Always check both cases when solving absolute value equations!

Use a graphing calculator to solve 5.7x = |2x + 1|.

Learn how to solve the absolute value equation 5.7x = |2x + 1| step by step. This problem covers important algebraic concepts, including absolute value properties and equation solving techniques, suitable for students in Grades 9-12.

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