To solve the integral \( I = \int_{-\frac{1}{2}}^{1} \left( |2x| + |x| \right) dx \), we can break it down into two parts based on the properties of the absolute value function.
### Step 1: Break the integral into parts
We can separate the integral into two parts:
\[
I = \int_{-\frac{1}{2}}^{1} |2x| \, dx + \int_{-\frac{1}{2}}^{1} |x| \, dx
\]
### Step 2: Evaluate \( \int_{-\frac{1}{2}}^{1} |2x| \, dx \)
We need to consider the intervals where the expression inside the absolute value changes:
- For \( x < 0 \) (i.e., from \( -\frac{1}{2} \) to \( 0 \)), \( |2x| = -2x \).
- For \( x \geq 0 \) (i.e., from \( 0 \) to \( 1 \)), \( |2x| = 2x \).
Thus, we can write:
\[
\int_{-\frac{1}{2}}^{1} |2x| \, dx = \int_{-\frac{1}{2}}^{0} (-2x) \, dx + \int_{0}^{1} (2x) \, dx
\]
Calculating each part:
1. For \( \int_{-\frac{1}{2}}^{0} (-2x) \, dx \):
\[
= -2 \left[ \frac{x^2}{2} \right]_{-\frac{1}{2}}^{0} = -2 \left( 0 - \frac{(-\frac{1}{2})^2}{2} \right) = -2 \left( 0 - \frac{1}{8} \right) = \frac{1}{4}
\]
2. For \( \int_{0}^{1} (2x) \, dx \):
\[
= 2 \left[ \frac{x^2}{2} \right]_{0}^{1} = 2 \left( \frac{1^2}{2} - 0 \right) = 2 \cdot \frac{1}{2} = 1
\]
Combining these results:
\[
\int_{-\frac{1}{2}}^{1} |2x| \, dx = \frac{1}{4} + 1 = \frac{5}{4}
\]
### Step 3: Evaluate \( \int_{-\frac{1}{2}}^{1} |x| \, dx \)
Similarly, we evaluate \( |x| \):
- For \( x < 0 \) (i.e., from \( -\frac{1}{2} \) to \( 0 \)), \( |x| = -x \).
- For \( x \geq 0 \) (i.e., from \( 0 \) to \( 1 \)), \( |x| = x \).
Thus, we can write:
\[
\int_{-\frac{1}{2}}^{1} |x| \, dx = \int_{-\frac{1}{2}}^{0} (-x) \, dx + \int_{0}^{1} (x) \, dx
\]
Calculating each part:
1. For \( \int_{-\frac{1}{2}}^{0} (-x) \, dx \):
\[
= -\left[ \frac{x^2}{2} \right]_{-\frac{1}{2}}^{0} = -\left( 0 - \frac{(-\frac{1}{2})^2}{2} \right) = \frac{1}{8}
\]
2. For \( \int_{0}^{1} (x) \, dx \):
\[
= \left[ \frac{x^2}{2} \right]_{0}^{1} = \frac{1^2}{2} - 0 = \frac{1}{2}
\]
Combining these results:
\[
\int_{-\frac{1}{2}}^{1} |x| \, dx = \frac{1}{8} + \frac{1}{2} = \frac{1}{8} + \frac{4}{8} = \frac{5}{8}
\]
### Step 4: Combine both integrals
Now we combine the results of both integrals:
\[
I = \frac{5}{4} + \frac{5}{8}
\]
To add these, we find a common denominator (which is 8):
\[
I = \frac{10}{8} + \frac{5}{8} = \frac{15}{8}
\]
### Step 5: Find \( 8I \)
Finally, we compute \( 8I \):
\[
8I = 8 \cdot \frac{15}{8} = 15
\]
Thus, the final answer is:
\[
\boxed{15}
\]
