The given expression is:

\[
Y = \frac{7 - 4x}{|2x + 5| + 3}
\]

### Key Points:

1. **Numerator**: \( 7 - 4x \) is linear.
2. **Denominator**: \( |2x + 5| + 3 \) involves an absolute value, so it's piecewise defined based on the sign of \( 2x + 5 \).

### Breaking down the absolute value:

The absolute value \( |2x + 5| \) can be written as a piecewise function:
- When \( 2x + 5 \geq 0 \), i.e., \( x \geq -\frac{5}{2} \), then \( |2x + 5| = 2x + 5 \).
- When \( 2x + 5 < 0 \), i.e., \( x < -\frac{5}{2} \), then \( |2x + 5| = -(2x + 5) = -2x - 5 \).

### So the function can be written as:

- **For \( x \geq -\frac{5}{2} \):**
  \[
  Y = \frac{7 - 4x}{(2x + 5) + 3} = \frac{7 - 4x}{2x + 8}
  \]
  
- **For \( x < -\frac{5}{2} \):**
  \[
  Y = \frac{7 - 4x}{(-2x - 5) + 3} = \frac{7 - 4x}{-2x - 2}
  \]

Would you like further details on how this affects the behavior of the function?

### Related Questions:
1. How does the function behave as \( x \to \infty \)?
2. What happens as \( x \to -\infty \)?
3. How does the function behave near \( x = -\frac{5}{2} \)?
4. What is the domain of the function?
5. Can the function be simplified further in specific cases?

**Tip:** Pay attention to absolute values as they create piecewise functions, which can change the behavior of a function based on different intervals of \(x\).

Explore the solution to the rational expression Y = (7 - 4x) / (|2x + 5| + 3), which involves the key concepts of absolute value and piecewise functions. Learn how to break down the function for different values of x. Suitable for Grades 10-12 students.

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