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You are asked to rewrite the absolute value function \( f(x) = |5x + 7| \) as a piecewise-defined function with linear parts. To achieve this, we will split the absolute value function into two cases based on when the expression inside the absolute value is positive or negative.

### Step-by-Step Solution:

The function \( f(x) = |5x + 7| \) can be written as:

1. **When \( 5x + 7 \geq 0 \) (i.e., \( x \geq -\frac{7}{5} \))**:
   In this case, the absolute value of \( 5x + 7 \) is just \( 5x + 7 \).
   So, for \( x \geq -\frac{7}{5} \), \( f(x) = 5x + 7 \).

2. **When \( 5x + 7 < 0 \) (i.e., \( x < -\frac{7}{5} \))**:
   In this case, the absolute value of \( 5x + 7 \) is \( -(5x + 7) = -5x - 7 \).
   So, for \( x < -\frac{7}{5} \), \( f(x) = -5x - 7 \).

### Piecewise-Defined Function:
\[
f(x) =
\begin{cases}
5x + 7 & \text{if } x \geq -\frac{7}{5} \\
-5x - 7 & \text{if } x < -\frac{7}{5}
\end{cases}
\]

This is the absolute value function rewritten as a piecewise-defined function.

Would you like any additional clarification, or do you have further questions on this?

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Here are 5 related questions to expand on this concept:

1. How do you graph the piecewise-defined function for \( f(x) = |5x + 7| \)?
2. Can you find the point where \( f(x) = |5x + 7| \) intersects the x-axis?
3. How would the piecewise function change if it was \( f(x) = |3x - 4| \) instead?
4. How do we solve \( |5x + 7| = 10 \)?
5. What is the domain and range of the function \( f(x) = |5x + 7| \)?

**Tip**: When dealing with absolute value functions, always consider breaking the function down into cases where the expression inside the absolute value is positive or negative.

Write the absolute value function as a piecewise-defined function with linear parts.

f(x) = |5x + 7|

Learn how to write the absolute value function f(x) = |5x + 7| as a piecewise-defined function with linear parts. This step-by-step solution explains the process and includes cases based on when the expression inside the absolute value is positive or negative. Suitable for Grades 9-10 students learning about piecewise and absolute value functions.

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