To solve the equation $\|x| + 5| = 4$, we need to consider the absolute values carefully.

First, we need to address the inner absolute value $|x|$. We know that $|x| \geq 0$ for any real number $x$. Let's analyze the outer absolute value expression $|x| + 5$:

1. $|x| + 5$ represents a non-negative number since both $|x|$ and 5 are non-negative.
2. The equation $|x| + 5 = 4$ suggests that $|x| + 5 = 4$ or $|x| + 5 = -4$.

However, since $|x| + 5$ is always non-negative, it can never be $-4$. Therefore, we only consider:

$|x| + 5 = 4$

Solving for $|x|$:

$|x| + 5 = 4$ $|x| = 4 - 5$ $|x| = -1$

Since $|x|$ is the absolute value of $x$ and must be non-negative, $|x| = -1$ has no solution because an absolute value cannot be negative.

Thus, there are no solutions to the equation $\|x| + 5| = 4$.

Do you want details or have any questions?

Here are 5 related questions:

1. How do you solve an equation with a single absolute value expression?
2. What are the properties of absolute values in equations?
3. Can absolute value equations have no solutions, and if so, why?
4. How do you solve $|x| = a$ for any positive $a$?
5. What happens when you graph $|x| + 5 = 4$?

Tip: When solving absolute value equations, always consider both the positive and negative scenarios, but remember that absolute values themselves cannot be negative.