Given the relationship \(|y + 3| = |x|\), let's analyze what happens when \( y = -4 \).

### Substitution:
Substitute \( y = -4 \) into the equation:

\[
|y + 3| = |x|
\]

\[
|-4 + 3| = |x|
\]

\[
|-1| = |x|
\]

\[
1 = |x|
\]

### Solution:
The equation \( |x| = 1 \) means that \( x \) can be either \( 1 \) or \( -1 \).

So, when \( y = -4 \), the corresponding values of \( x \) are \( x = 1 \) or \( x = -1 \).

### Explanation:
The absolute value allows for both positive and negative values of \( x \), maintaining the relationship between \( x \) and \( y \) regardless of their signs. In this case, even though \( y = -4 \) is less than the original \( y = -2 \), the absolute value relationship ensures that \( x \) can still be either \( 1 \) or \( -1 \) without violating the equation.

### Would you like more details, or do you have any questions?

Here are five related questions for further exploration:
1. How do absolute values affect the range of solutions in an equation?
2. What happens if you substitute \( y = 0 \) in the equation \(|y + 3| = |x|\)?
3. How would you graph the equation \(|y + 3| = |x|\) on a coordinate plane?
4. Can there be multiple solutions for \( x \) given a value of \( y \) in absolute value equations?
5. How does the relationship between \( x \) and \( y \) change if the initial conditions are altered?

**Tip:** Always remember that absolute values split an equation into two cases: positive and negative. This is essential in solving equations with absolute values.

Explore the absolute value equation |y + 3| = |x| with detailed step-by-step analysis. Understand how absolute values affect equations, solve for multiple values of x when y = -4, and learn about the relationship between x and y in absolute value equations. Suitable for Grades 9-12.

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