Given that k is an integer and the equation |x - k| + |x - 5| = 9 has infinite solutions, find the sum of the maximum and minimum values of k.

To find the sum of the maximum and minimum values of k, we need to analyze the given equation |x - k| + |x - 5| = 9.

Step 1: Analyzing the absolute value expressions The equation involves two absolute value expressions: |x - k| and |x - 5|. Let's consider the different cases based on the signs of these expressions.

Case 1: x - k ≥ 0 and x - 5 ≥ 0 In this case, both absolute value expressions are positive, so we can rewrite the equation as (x - k) + (x - 5) = 9.

Simplifying this equation, we get: 2x - k - 5 = 9 2x - k = 14

Case 2: x - k < 0 and x - 5 ≥ 0 In this case, the first absolute value expression is negative, so we need to negate it. The equation becomes (-x + k) + (x - 5) = 9.

Simplifying this equation, we get: -k - 5 = 9 -k = 14

Case 3: x - k ≥ 0 and x - 5 < 0 In this case, the second absolute value expression is negative, so we need to negate it. The equation becomes (x - k) + (-x + 5) = 9.

Simplifying this equation, we get: -k + 5 = 9 -k = 4

Case 4: x - k < 0 and x - 5 < 0 In this case, both absolute value expressions are negative, so we can rewrite the equation as (-x + k) + (-x + 5) = 9.

Simplifying this equation, we get: -2x + k + 5 = 9 -2x + k = 4

Step 2: Analyzing the solutions Since the equation |x - k| + |x - 5| = 9 has infinite solutions, it means that all the cases we analyzed above must hold true simultaneously. This implies that the equations obtained in each case must be consistent.

Comparing the equations from Case 2 and Case 4, we can see that they are the same: -k = 14 and -2x + k = 4. Therefore, we can conclude that the value of k is 14.

Step 3: Finding the sum of the maximum and minimum values of k Since we have determined that k = 14, the sum of the maximum and minimum values of k is 14 + 14 = 28.

Therefore, the sum of the maximum and minimum values of k is 28.