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The problem asks to rewrite the absolute value function \( f(x) = |5x + 4| \) as a piecewise-defined function. The key to solving this is to break it into two parts, based on the definition of absolute value:

\[
|a| = \begin{cases} 
a & \text{if } a \geq 0 \\
-a & \text{if } a < 0 
\end{cases}
\]

For \( f(x) = |5x + 4| \):

1. The expression \( 5x + 4 \) is positive or zero when \( 5x + 4 \geq 0 \), i.e., \( x \geq -\frac{4}{5} \). In this case, \( f(x) = 5x + 4 \).
2. When \( 5x + 4 < 0 \), which means \( x < -\frac{4}{5} \), the absolute value of \( 5x + 4 \) becomes \( -(5x + 4) \).

Thus, the piecewise function is:

\[
f(x) = \begin{cases} 
5x + 4 & \text{if } x \geq -\frac{4}{5} \\
-(5x + 4) & \text{if } x < -\frac{4}{5}
\end{cases}
\]

This matches the structure shown in the boxes on your screen, where you can fill in:

\[
f(x) = \begin{cases} 
5x + 4 & \text{if } x \geq -\frac{4}{5} \\
-(5x + 4) & \text{if } x < -\frac{4}{5}
\end{cases}
\]

Do you need more details or have any further questions?

Here are five related questions to explore:
1. How do you handle the absolute value of more complex functions?
2. What happens when a function inside the absolute value has multiple roots?
3. How would you graph a piecewise function like this one?
4. What is the significance of the point \( x = -\frac{4}{5} \)?
5. How do you integrate piecewise functions?

**Tip:** Always identify where the expression inside the absolute value changes sign to set up your piecewise function correctly!

Write the absolute value function f(x) = |5x + 4| as a piecewise-defined function with linear parts.

Learn how to rewrite the absolute value function f(x) = |5x + 4| as a piecewise-defined function with linear parts. This problem introduces piecewise functions and explores the concept of absolute value. Suitable for students in Grades 8-10.

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