`int_(-3)^(3) (x-4)/((|x-4|))dx`=

Options

A. 0
B. 6
C. `-6`
D. none

Answer

To solve the integral

\[ \int_{-3}^{3} \frac{x-4}{|x-4|} \, dx, \]

we need to analyze the expression \(\frac{x-4}{|x-4|}\).

### Step 1: Determine the behavior of \(|x-4|\)

The absolute value function \(|x-4|\) can be expressed piecewise:

- For \(x < 4\), \(|x-4| = -(x-4) = 4-x\)
- For \(x \geq 4\), \(|x-4| = x-4\)

Since our limits of integration are from \(-3\) to \(3\), which are both less than \(4\), we can use the first case:

\[ |x-4| = 4 - x \quad \text{for } x \in [-3, 3]. \]

### Step 2: Rewrite the integrand

Now we can rewrite the integrand:

\[ \frac{x-4}{|x-4|} = \frac{x-4}{4-x}. \]

Since \(4-x\) is positive for \(x < 4\), we can simplify:

\[ \frac{x-4}{4-x} = -1. \]

### Step 3: Set up the integral

Now we can set up the integral:

\[ \int_{-3}^{3} -1 \, dx. \]

### Step 4: Calculate the integral

The integral of \(-1\) over the interval from \(-3\) to \(3\) is:

\[ -1 \cdot (3 - (-3)) = -1 \cdot 6 = -6. \]

### Final Answer

Thus, the value of the integral is:

\[ \int_{-3}^{3} \frac{x-4}{|x-4|} \, dx = -6. \]