Given that k is an integer and the equation |x - k| + |x + 3| = 12 has infinite solutions, find the sum of the maximum and minimum values of k.

Sure, let's solve the equation step by step.

Step 1: Rewrite the equation using the definition of absolute value. |x - k| + |x + 3| = 12

Step 2: Consider the different cases for the absolute value expressions.

Case 1: (x - k) ≥ 0 and (x + 3) ≥ 0 In this case, the absolute value expressions can be simplified to: (x - k) + (x + 3) = 12 2x - k + 3 = 12 2x - k = 9

Case 2: (x - k) ≥ 0 and (x + 3) < 0 In this case, the absolute value expressions can be simplified to: (x - k) - (x + 3) = 12 x - k - x - 3 = 12

• k - 3 = 12
• k = 15

Case 3: (x - k) < 0 and (x + 3) ≥ 0 In this case, the absolute value expressions can be simplified to: -(x - k) + (x + 3) = 12 -k + x + x + 3 = 12 2x - k + 3 = 12 2x - k = 9

Case 4: (x - k) < 0 and (x + 3) < 0 In this case, the absolute value expressions can be simplified to: -(x - k) - (x + 3) = 12 -k + x - x - 3 = 12

• k - 3 = 12
• k = 15

Step 3: Solve each case separately.

Case 1: 2x - k = 9 Case 2: - k = 15 Case 3: 2x - k = 9 Case 4: - k = 15

Step 4: Find the maximum and minimum values of k.

The maximum value of k is the smallest negative value, which is -15. The minimum value of k is the largest positive value, which is 9.

Step 5: Calculate the sum of the maximum and minimum values of k.

Sum of maximum and minimum values of k = -15 + 9 = -6

Therefore, the sum of the maximum and minimum values of k is -6.