When is `|x-4|` equal to `4-x`?

Answer

To solve the equation \( |x - 4| = 4 - x \), we need to consider the definition of the absolute value and break the problem into cases based on the value of \( x \).

### Step 1: Understand the Absolute Value
The absolute value \( |x - 4| \) can be defined as:
- \( x - 4 \) when \( x - 4 \geq 0 \) (i.e., \( x \geq 4 \))
- \( -(x - 4) \) when \( x - 4 < 0 \) (i.e., \( x < 4 \))

### Step 2: Case 1 - When \( x \geq 4 \)
In this case, we have:
\[ |x - 4| = x - 4 \]
So, the equation becomes:
\[ x - 4 = 4 - x \]
Now, we can solve for \( x \):
\[ x - 4 + x = 4 \]
\[ 2x - 4 = 4 \]
\[ 2x = 8 \]
\[ x = 4 \]
Since \( x = 4 \) is within the case condition \( x \geq 4 \), this solution is valid.

### Step 3: Case 2 - When \( x < 4 \)
In this case, we have:
\[ |x - 4| = -(x - 4) = 4 - x \]
So, the equation becomes:
\[ -(x - 4) = 4 - x \]
This simplifies to:
\[ 4 - x = 4 - x \]
This equation is always true for any \( x < 4 \). Therefore, all values of \( x \) that are less than \( 4 \) satisfy the equation.

### Final Solution
Combining both cases, we find that:
- For \( x \geq 4 \), the solution is \( x = 4 \).
- For \( x < 4 \), all values satisfy the equation.

Thus, the final answer is:
\[ x \leq 4 \]