The task involves determining the moment of inertia $I_x$ of a structure with respect to the $x$-axis, given its geometric shape described by the equations $y = 2x$ and $y = -2x$, within the range $|x| \leq 1$. The diagram suggests a symmetric, double-triangle shape centered around the origin.

Step 1: Determine the Geometry

The area described consists of two identical triangles with vertices at (0,0), (1,2), and (-1,-2). The area of one of these triangles is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2$ Hence, the total area of both triangles is $4$.

Step 2: Moment of Inertia about the x-axis ($I_x$)

The moment of inertia of a triangle about an axis through its base (parallel to the base) is given by: $I = \frac{1}{36} \times \text{base} \times \text{height}^3$ For one triangle, using $\text{base} = 2$ and $\text{height} = 2$: $I = \frac{1}{36} \times 2 \times 8 = \frac{4}{9}$ Since the axis passes through the centroid (central axis for the whole structure), there is no need to use the parallel axis theorem. The total $I_x$ for both triangles is: $I_x = 2 \times \frac{4}{9} = \frac{8}{9}$

Step 3: Adjust for the Shift in Axis

When the reference axis is moved from $(0,0)$ to $(0, -2)$, we apply the parallel axis theorem: $I'_x = I_x + \text{Area} \times d^2$ where $d = 2$ is the vertical distance the axis has moved: $I'_x = \frac{8}{9} + 4 \times 4 = \frac{8}{9} + 16 = \frac{152}{9}$

Therefore, the moment of inertia $I'_x$ of the structure about the new axis $y = -2$ is $\frac{152}{9}$.

Do you need further details on any of these steps or have other questions? Here are some related questions you might consider:

1. How would $I_x$ change if the shape was reflected over the y-axis?
2. What would be the moment of inertia around the y-axis for the original position?
3. How does the moment of inertia affect the structural stability of the object?
4. What is the parallel axis theorem and how is it applied in various contexts?
5. What are other geometric transformations that might affect the moment of inertia?

Tip: When calculating moments of inertia, always verify the axis about which the inertia is calculated, as shifting the axis can significantly affect the results.