Step-by-Step Solution
**Step 1: Understand the absolute value function**
The absolute value function is defined as:
\[ |x| = \begin{cases}
-x, & \text{if } x < 0 \\
x, & \text{if } x \ge 0
\end{cases} \]
**Step 2: Rewrite the integral using the definition of |x|**
Since the integral is from -1 to 1, we need to split the integral at x = 0:
\[ \int_{-1}^{1} \frac{|x|}{x} \, dx = \int_{-1}^{0} \frac{|x|}{x} \, dx + \int_{0}^{1} \frac{|x|}{x} \, dx \]
For \(x < 0\), \(|x| = -x\), and for \(x > 0\), \(|x| = x\). Therefore:
\[ \int_{-1}^{0} \frac{-x}{x} \, dx + \int_{0}^{1} \frac{x}{x} \, dx = \int_{-1}^{0} -1 \, dx + \int_{0}^{1} 1 \, dx \]
**Step 3: Evaluate the integrals**
\[ \int_{-1}^{0} -1 \, dx = -x \Big|_{-1}^{0} = -(0 - (-1)) = -1 \]
\[ \int_{0}^{1} 1 \, dx = x \Big|_{0}^{1} = (1 - 0) = 1 \]
**Step 4: Add the results**
\[ -1 + 1 = 0 \]