The given set $\{(x, y) \mid |x + y| = 2\}$ represents all the points $(x, y)$ in the coordinate plane that satisfy the absolute value equation $|x + y| = 2$.

Analyzing the Equation

The absolute value equation $|x + y| = 2$ can be interpreted in two separate cases, because the absolute value function outputs the non-negative value of its argument:

1. Case 1: $x + y = 2$
2. Case 2: $x + y = -2$

Case 1: $x + y = 2$

This equation represents a straight line in the coordinate plane with a slope of $-1$ and a y-intercept of $2$. The line passes through the points $(0, 2)$ and $(2, 0)$.

Case 2: $x + y = -2$

Similarly, this equation represents another straight line in the coordinate plane with a slope of $-1$ but with a y-intercept of $-2$. This line passes through the points $(0, -2)$ and $(-2, 0)$.

Geometric Interpretation

The solution set $\{(x, y) \mid |x + y| = 2\}$ consists of all the points on the two lines:

• Line 1: $x + y = 2$
• Line 2: $x + y = -2$

These lines are parallel to each other and have a slope of $-1$. They are two units apart vertically in the coordinate plane.

Graphical Representation

To visualize, the set forms a pair of parallel lines on the Cartesian plane:

• One line goes through the points $(0, 2)$ and $(2, 0)$.
• The other line goes through the points $(0, -2)$ and $(-2, 0)$.

Summary

The set $\{(x, y) \mid |x + y| = 2\}$ represents the union of the two lines $x + y = 2$ and $x + y = -2$. All points $(x, y)$ lying on either of these two lines are part of the set.

Would you like further details or have any questions about this topic?

Here are some related questions you might be interested in:

1. How do you graph the absolute value equation $|x - y| = 3$?
2. What are the solutions to the equation $|x + 2y| = 4$?
3. How does the absolute value function affect linear equations?
4. How can we find the intersection points of the lines $x + y = 2$ and $x + y = -2$ with other lines?
5. What is the distance between two parallel lines in general?

Tip: When solving absolute value equations, remember to consider both the positive and negative scenarios to find all possible solutions.