A set of numbers contains 7 integers and has an average

for a set of 7 numbers if the median is 23 then we can have 2 cases

1) all numbers are 23 { 23,23,23,23,23,23,23}, in this case too average is 23

2) {# <= 23, # <= 23, # <= 23, 23( center digit equal to 23) , # >= 23, >= 23, >= 23}, median of 7 numbers is 23 and average is 23, means that half of the numbers are below or equal to the median and half of the numbers are equal to or above the median .

Now the given condition in the question is that the maximum = 4X + 15 , where x is the smallest one .So we know that all numbers are not equal .

now in order to maximize the range we have to keep the other 6 as small as possible .

{ x,x,x,23,23,23,4X+15}

now if the smallest is x then largest is 4x+15, this is given, so we cannot change these two.
the center has to be 23 as this is the median.

so the smallest value for the 2, and 3, digit has to be equal to the smallest one which is equal to x , we cannot take anything larger then x as that could make the range smaller , but want to find the largest range , so we have to take the smallest numbers.

similarly:

the smallest value for 4th and 5th digit cannot be smaller than 23, as 23 is the median , and cannot be larger the 23 as again we are looking for the largest range, so we have the keep the 4th and 5th digits as small as possible , and the smallest we can get for the 4th and 5th digit is 23.

hence Bunuel's Statement

Now, in order to maximize the range we need to make the second and the third numbers equal to x and the fifth and sixth numbers equal to 23, so the set should be {x, x, x, 23, 23, 23, 4x+15}.

Hope it is clear to you now .Please ask if any doubt remains .