Question

The number of pairs \( (x, y) \) of integers satisfying the inequality \( |x - 5| + |y - 5| \leq 6 \) is ____________.

Hint:

To solve inequalities with absolute values, break the absolute value terms into cases, and then count the possible integer solutions for each case. This helps visualize the number of valid solutions.

Answer 1

Step 1: Rewrite the inequality. The given inequality is: \[ |x - 5| + |y - 5| \leq 6 \] This inequality represents a diamond-shaped region in the coordinate plane, where the center of the diamond is at \( (5, 5) \), and the total "distance" from the center to any point inside the diamond is 6. 

Step 2: Analyze the constraints on \( x \) and \( y \). We have: \[ |x - 5| + |y - 5| \leq 6 \] This inequality can be broken down as follows: - The value \( |x - 5| \) is the horizontal distance from \( x \) to 5, and \( |y - 5| \) is the vertical distance from \( y \) to 5. - The sum of these distances must be at most 6, which means both \( x \) and \( y \) must stay within a "range" from 5, based on the condition \( |x - 5| + |y - 5| \leq 6 \). 

Step 3: Count the possible integer pairs \( (x, y) \). - For \( x = 5 \), the inequality becomes \( |y - 5| \leq 6 \), so \( y \) can take any integer value from \( -1 \) to \( 11 \), giving 13 values for \( y \).
- For \( x = 4 \) or \( x = 6 \), the inequality becomes \( |y - 5| \leq 5 \), so \( y \) can take any integer value from \( 0 \) to \( 10 \), giving 11 values for \( y \) for each of these \( x \)-values.
- For \( x = 3 \) or \( x = 7 \), the inequality becomes \( |y - 5| \leq 4 \), so \( y \) can take any integer value from \( 1 \) to \( 9 \), giving 9 values for \( y \) for each of these \( x \)-values.
- For \( x = 2 \) or \( x = 8 \), the inequality becomes \( |y - 5| \leq 3 \), so \( y \) can take any integer value from \( 2 \) to \( 8 \), giving 7 values for \( y \) for each of these \( x \)-values.
- For \( x = 1 \) or \( x = 9 \), the inequality becomes \( |y - 5| \leq 2 \), so \( y \) can take any integer value from \( 3 \) to \( 7 \), giving 5 values for \( y \) for each of these \( x \)-values.
- For \( x = 0 \) or \( x = 10 \), the inequality becomes \( |y - 5| \leq 1 \), so \( y \) can take any integer value from \( 4 \) to \( 6 \), giving 3 values for \( y \) for each of these \( x \)-values.
- For \( x = -1 \) or \( x = 11 \), the inequality becomes \( |y - 5| \leq 0 \), so \( y = 5 \), giving 1 value for \( y \) for each of these \( x \)-values. 

Step 4: Calculate the total number of pai\rupee Now, add the total number of possible pairs for each \( x \)-value: \[ 13 + 2(11) + 2(9) + 2(7) + 2(5) + 2(3) + 2(1) = 13 + 22 + 18 + 14 + 10 + 6 + 2 = 85 \] Thus, the number of pairs \( (x, y) \) of integers satisfying the inequality is: \[ \boxed{85} \]