Problem:
Let [z] denote the greatest integer not exceedng z. Let x and y satisfy the simultaneous equations
​y=2[x]+3y=3[x−2]+5.​
If x is not an integer, then x+y is
Answer Choices:
A. an integer
B. between 4 and 5
C. between −4 and 4
D. between 15 and 16
E. 16.5
Solution:
We have
2[x]+32[x]+3[x]​=3[x−2]+5=3([x]−2)+5=4​
Therefore, 4<x<5, and y=2[x]+3=11. Hence, 15<x+y<16. (Alternately, if one draws the graphs of y=2[x]+3 and y=3[x−2]+5 one can see that they overlap when 4<x<5).