\( 0 \leq \frac { \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } + x + y + 1 } { | x + y | } \leq \frac { 1 } { \sqrt { 2 } } \frac { \sqrt { ( 1 + 2 x ) ^ { 2 } + ( 1 + 2 y ) ^ { 2 } } + x + y + 1 } { | x + y | } \)

To analyze the inequality \[ 0 \leq \frac { \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } + x + y + 1 } { | x + y | } \leq \frac { 1 } { \sqrt { 2 } } \frac { \sqrt { ( 1 + 2 x ) ^ { 2 } + ( 1 + 2 y ) ^ { 2 } } + x + y + 1 } { | x + y | }, \] we will break it down into two parts: the left-hand side and the right-hand side. ### Left-hand side analysis The left-hand side of the inequality is \[ 0 \leq \frac { \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } + x + y + 1 } { | x + y | }. \] This expression is non-negative if the numerator is non-negative. The numerator is \[ \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } + x + y + 1. \] 1. **Case 1**: If \( x + y = 0 \), then the expression is undefined since we cannot divide by zero. 2. **Case 2**: If \( x + y \neq 0 \), we need to check when the numerator is non-negative. The term \( \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } \) is always non-negative. The terms \( x + y + 1 \) can be analyzed based on the values of \( x \) and \( y \). ### Right-hand side analysis The right-hand side of the inequality is \[ \frac { \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } + x + y + 1 } { | x + y | } \leq \frac { 1 } { \sqrt { 2 } } \frac { \sqrt { ( 1 + 2 x ) ^ { 2 } + ( 1 + 2 y ) ^ { 2 } } + x + y + 1 } { | x + y | }. \] To analyze this, we can rewrite it as: \[ \sqrt { | ( 1 + 2 x ) ( 1 + 2 y ) | } + x + y + 1 \leq \frac { 1 } { \sqrt { 2 } } \left( \sqrt { ( 1 + 2 x ) ^ { 2 } + ( 1 + 2 y ) ^ { 2 } } + x + y + 1 \right). \] This inequality suggests that we need to compare the left-hand side with a scaled version of the right-hand side. ### Conclusion To fully analyze the inequality, we would need to consider specific values of \( x \) and \( y \) to see if the inequality holds. 1. **When \( x = 0 \) and \( y = 0 \)**, both sides equal zero, satisfying the inequality. 2. **When \( x = 1 \) and \( y = 0 \)**, we can compute both sides to check if the inequality holds. 3. **When \( x = -1 \) and \( y = 0 \)**, we can also compute both sides. In general, the inequality will hold under certain conditions on \( x \) and \( y \). The analysis can be extended to find specific ranges or conditions under which the inequality is satisfied. To summarize, the inequality is valid under certain conditions, and further exploration with specific values or ranges for \( x \) and \( y \) would be necessary to fully characterize the solution set.