https://file.aiquickdraw.com/mathbot/file/172630453894116yglwkc.jpg

The expression you uploaded is:

\[
\frac{|x - 1| - |-2y|}{|3y + 2x|}
\]

This expression involves absolute values. Let's break it down:

### Numerator:
- \( |x - 1| \) is the absolute value of \( x - 1 \).
- \( |-2y| = 2|y| \) because the absolute value of \( -2y \) is \( 2 \times |y| \).

So the numerator simplifies to:
\[
|x - 1| - 2|y|
\]

### Denominator:
- \( |3y + 2x| \) remains as is because it is the absolute value of the linear combination \( 3y + 2x \).

Thus, the entire expression simplifies to:

\[
\frac{|x - 1| - 2|y|}{|3y + 2x|}
\]

Let me know if you need further simplifications or details, and feel free to ask any clarifying questions!

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Here are five related questions that might expand on this:

1. How do you solve equations involving absolute values?
2. What is the geometric interpretation of absolute value functions?
3. How do we handle cases when the expression inside an absolute value equals zero?
4. Can you graph this function to understand its behavior?
5. How do absolute values affect the domain and range of this expression?

**Tip**: When simplifying absolute value expressions, consider the sign of the terms inside the absolute values, as they could affect the overall result!

Simplify the expression \(\frac{\|x - 1\| - \|-2y\|}{\|3y + 2x\|}\).

Learn how to simplify the expression |x - 1| - |-2y| / |3y + 2x| using absolute value properties and algebraic techniques. This problem covers key algebraic concepts, such as simplification of absolute values and linear expressions, suitable for students in Grades 8-10.

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