The given equation is:

$|x_1| + 2|x_2| = 8$

Step 1: Consider Different Cases for Absolute Values

Since we have absolute values, we need to analyze all possible cases for $x_1$ and $x_2$:

1. Case 1: $x_1 \geq 0$ and $x_2 \geq 0$
   $x_1 + 2x_2 = 8$

2. Case 2: $x_1 \geq 0$ and $x_2 \leq 0$
   $x_1 + 2(-x_2) = 8 \quad \Rightarrow \quad x_1 - 2x_2 = 8$

3. Case 3: $x_1 \leq 0$ and $x_2 \geq 0$
   $(-x_1) + 2x_2 = 8 \quad \Rightarrow \quad -x_1 + 2x_2 = 8$

4. Case 4: $x_1 \leq 0$ and $x_2 \leq 0$
   $(-x_1) + 2(-x_2) = 8 \quad \Rightarrow \quad -x_1 - 2x_2 = 8$

Step 2: Analyze Solutions

Each of these cases represents a linear equation that describes a region in the $(x_1, x_2)$ plane. The solution set consists of all points that satisfy at least one of these equations.

Would you like me to solve for specific values of $x_1$ or $x_2$, or would you like a graphical representation of the solution set?

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Here are five related questions to expand your understanding:

1. How do absolute value equations affect the solution sets geometrically?
2. Can this equation represent a bounded region in the coordinate plane?
3. How would the solution change if the equation were $|x_1| + 2|x_2| \leq 8$?
4. What is the relationship between absolute value equations and piecewise functions?
5. How can we extend this concept to higher dimensions?

Tip: Always break absolute value equations into separate cases and solve each one carefully to ensure you capture all solutions.