Question 1058559
Let x represent one atomic number, and let y represent the other.


Since the sum is not more than 22, you can write:


x + y ≤ 22


Since the average is equal to 11, you can write:


(x + y)/2 = 11


which can be rearranged as:


x + y = 2(11)


x + y = 22


Thus, the sum is not only "not more than 22", is must be exactly 22.


Then, since the difference is less than 16, you can write:


x - y < 16


Solving x + y = 22 for y gives:


y = 22 - x


Substituting that into the inequality gives:


x - (22 - x) < 16


x - 22 + x < 16


2x - 22 < 16


2x < 16 + 22


2x < 38


x < 19


Then, solving x + y = 22 for x gives:


x = 22 - y


Substituting this into inequality gives:


22 - y - y < 16


22 - 2y < 16


-2y < 16 - 22


-2y < -6


y > 3  (remember to reverse the inequality sign when multiplying or dividing by a negative number)


Thus, the combinations must have x < 19 and y > 3


The set of combinations is then:


(4, 18), (5, 17), (6, 16), (7, 15), (8, 14), (9, 13), (10, 12), and (11, 11)