Lets consider a problem that involves hyperplanes in two dimensions 
(1) Sketch the hyperplane 1 + 3X1 - X2 = 0. Indicate the set of points for which 1 + 3X1 - X2 > 0, as well as the set of points for which 1 + 3X1 - X2 < 0.
(2) On the same plot, sketch the hyperplane -2 + X1 + 2X2 = 0. Indicate the set of points for which -2 + X1 + 2X2 > 0, as well as the set of the points for which -2 + X1 + 2X2 < 0.

Here is the plot representing the two hyperplanes and their respective regions:

1. **Hyperplane 1 (blue line):** \( 1 + 3X_1 - X_2 = 0 \)
   - The region where \( 1 + 3X_1 - X_2 > 0 \) is shaded in light red.
   - The region where \( 1 + 3X_1 - X_2 < 0 \) is shaded in light blue.

2. **Hyperplane 2 (green line):** \( -2 + X_1 + 2X_2 = 0 \)
   - The region where \( -2 + X_1 + 2X_2 > 0 \) is shaded in light yellow.
   - The region where \( -2 + X_1 + 2X_2 < 0 \) is shaded in light green.

Would you like more details about the equations, or do you have any questions?

Here are 5 related questions to expand the topic:
1. How do the normal vectors of these hyperplanes relate to their orientation?
2. Can the intersection of the two hyperplanes be found algebraically?
3. How do you classify the regions formed by these hyperplanes geometrically?
4. How would these hyperplanes look in higher dimensions (e.g., 3D)?
5. What is the general form of a hyperplane in n-dimensional space?

**Tip:** Hyperplanes in 2D are just lines, but in higher dimensions, they separate the space into distinct half-spaces.

Lets consider a problem that involves hyperplanes in two dimensions: (1) Sketch the hyperplane 1 + 3X1 - X2 = 0. Indicate the set of points for which 1 + 3X1 - X2 > 0, as well as the set of points for which 1 + 3X1 - X2 < 0. (2) On the same plot, sketch the hyperplane -2 + X1 + 2X2 = 0. Indicate the set of points for which -2 + X1 + 2X2 > 0, as well as the set of points for which -2 + X1 + 2X2 < 0.

Explore how to sketch and analyze two hyperplanes in two dimensions: 1 + 3X1 - X2 = 0 and -2 + X1 + 2X2 = 0. Learn about hyperplanes, inequalities, and half-space division in this detailed solution, suitable for undergraduate-level students studying linear algebra.

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