Solution: We analyze the inequality $ |x + y| + |x - y| \leq 8 $. This expression is symmetric and represents a diamond (rhombus) centered at the origin.

["Solving the Inequality ( |x + y| + |x - y| \leq 8 ): Understanding the Diamond Shape in the Coordinate Plane", "The expression ( |x + y| + |x - y| \leq 8 ) presents a fascinating geometric interpretation rooted in absolute values and symmetry. Rather than treating it merely as an algebraic inequality, we can uncover the rich geometric meaning behind it—a symmetric diamond (rhombus)—centered at the origin ((0, 0)) in the coordinate plane.", "---", "### Understanding the Expression", "To analyze ( |x + y| + |x - y| \leq 8 ), we begin by recalling that absolute values represent distances from zero on a number line. This inequality combines two absolute expressions involving linear combinations of (x) and (y).", "Let’s denote:\n- ( u = x + y )\n- ( v = x - y )", "Then the inequality becomes:\n[\n|u| + |v| \leq 8\n]", "This is the standard form of a diamond (rhombus) in the (uv)-plane centered at the origin with vertices at ( (8,0), (-8,0), (0,8), (0,-8) ).", "---", "### Mapping Back to the (xy)-Plane", "Since ( u = x + y ) and ( v = x - y ), we perform a coordinate transformation to interpret the region in the original (xy)-plane.", "The transformation equations are:\n[\nu = x + y, \quad v = x - y\n]\nSolving for (x) and (y):\n[\nx = \frac{u + v}{2}, \quad y = \frac{u - v}{2}\n]", "This is a linear transformation with determinant:\n[\n\ ext{det} = \begin{vmatrix} \frac{1}{2} & \frac{1}{2} \ \frac{1}{2} & -\frac{1}{2} \end{vmatrix} = -\frac{1}{2} - \frac{1}{2} = -1 \quad \Rightarrow \quad |\ ext{det}| = 1\n]\nSince the absolute value is 1, the transformation preserves area (orientation-reversing but area-preserving).", "The original inequality ( |u| + |v| \leq 8 ) defines a diamond in the (uv)-plane with side length ( 8\sqrt{2} ). Because of the area-preserving linear transformation, the image in the (xy)-plane is also a diamond—same shape but rotated and oriented differently.", "---", "### Identifying the Diamond’s Vertices and Sides", "In the (uv)-plane, the vertices of ( |u| + |v| \leq 8 ) occur where one variable is zero:", "- When ( u = 8, v = 0 \Rightarrow x = \frac{8+0}{2} = 4,\ y = \frac{8-0}{2} = 4 \Rightarrow (4, 4) )\n- When ( u = -8, v = 0 \Rightarrow (-4, -4) )\n- When ( u = 0, v = 8 \Rightarrow x = \frac{0+8}{2} = 4,\ y = \frac{0-8}{2} = -4 \Rightarrow (4, -4) )\n- When ( u = 0, v = -8 \Rightarrow (-4, 4) )", "So the diamond has vertices at:\n[\n(4, 4),\ (4, -4),\ (-4, -4),\ (-4, 4)\n]\nThis forms a rhombus symmetric about both axes, centered at the origin, with diagonals aligned along the lines ( y = x ) and ( y = -x ).", "---", "### Geometric Interpretation and Properties", "The expression ( |x + y| + |x - y| ) measures the sum of the absolute values of two dot products or projections:", "- ( |x + y| = |x \cdot 1 + y \cdot 1| ): magnitude of projection along vector ( \langle 1, 1 \rangle )\n- ( |x - y| = |x \cdot 1 + y \cdot (-1)| ): magnitude of projection along ( \langle 1, -1 \rangle )", "Their sum remains bounded, constraining points equidistant in this symmetric setup.", "Geometrically, the region satisfies:\n- Symmetry across both axes and the lines ( y = x ) and ( y = -x )\n- Horizontal and vertical "edges" tilted at 45°, forming diamond corners\n- All points lie within a diamond whose main diagonals span from ((4,4)) to ((-4,-4)) (length ( 8\sqrt{2} )) and ((4,-4)) to ((-4,4)) (also length ( 8\sqrt{2} ))", "---", "### Focusing on Key Features", "- Diagonal lengths: The full diagonal lengths are ( 8\sqrt{2} ) along ( y = x ) and ( y = -x ), shrinking to zero at the origin.\n- Edges: The sides connect the vertices as line segments forming a rotated square.\n- Boundary: The boundary ( |x + y| + |x - y| = 8 ) traces the edges of this diamond.", "---", "### Why This Diamonds Shape Matters", "Such symmetry simplifies analysis in multiple domains:\n- Geometry and symmetry studies\n- Optimization problems involving Manhattan or ( L_1 )-norm constraints\n- Linear algebra, due to preserved shape under orthogonal transformations\n- Computer graphics and robotics, where symmetric regions can define motion constraints or search spaces", "---", "### Visualizing the Inequality Region", "Plotting ( |x + y| + |x - y| \leq 8 ) reveals a diamonld centered at the origin with sharp corners at ((4, 4)), ((-4, -4)), ((4, -4)), and ((-4, 4)). This rhombus is entirely contained within the square defined by ( |x| + |y| \leq 8 ), but rotated 45°, confirming geometric consistency.", "---", "### Conclusion", "The inequality ( |x + y| + |x - y| \leq 8 ) describes a symmetric diamond (rhombus) in the plane, aligned with the diagonals, centered at the origin. Its vertices lie at the corners ((4, 4)), ((-4, -4)), ((4, -4)), and ((-4, 4)), illustrating how absolute value expressions encode geometric shapes through symmetry. This elegant solution connects algebra and geometry, making complex regions intuitive and analyzable through familiar diamond symmetry.", "Understanding such regions deepens insight into the interplay between equations and shapes—key in advanced mathematics, engineering applications, and computational geometry.", "---", "Keywords: ( |x + y| + |x - y| \leq 8 ), diamond shape, rhombus geometry, absolute value inequality, coordinate geometry, symmetry, linear transformation, L1 norm region, coordinate plane."]