Problem:
If x and y are non-zero real numbers such that
∣x∣+y=3 and ∣x∣y+x3=0
then the integer nearest to x−y is
Answer Choices:
A. −3
B. −1
C. 2
D. 3
E. 5
Solution:
If x>0, then x+y=3 and y+x2=0. Eliminate y from these simultaneous equations to obtain x2−x+3=0, which has no real roots. If x<0, then we have −x+y=3 and −y+x2=0, which have a simultaneous real solution, so x−y=−3.
Note. One need not obtain the solution, (x,y)=(21−13​​,27−13​​), to find the answer.
OR
Sketch y=3−∣x∣ and y=∣x∣−x3​={x2−x2​if x<0if x>0​.
Note that the graphs cross only on the half-line y=x+3,x<0. Therefore x−y=−3.
