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.