SOLUTION: The sum of the atomic number of two elements is not more than 22, with an average 11 and with a difference less than 16. What are the possible atomic numbers?

Question 1058559: The sum of the atomic number of two elements is not more than 22, with an average 11 and with a difference less than 16. What are the possible atomic numbers?
Answer by solve_for_x(190) (Show Source): You can put this solution on YOUR website!
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)
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