inequalities topic

Question

if x and y are the integers such that 0<x+y<10 then which of the following is true

 

(a) xy<2

(b)x>3

(c)x>4

(d)x+y=5

(e)x-y<8

Answer 1

The boundaries of the feasible region are the two lines:

 

y=-x

 

y=-x+10

 

All the solutions must lie above the first line and below the second line.

 

(a) Consider xy<2

 

If x<0, then both equations show that y>0, so xy<0<2.

 

If x>10, then both equation show that y<0, so xy<2.

 

When 0<x<10, the greatest value for xy is obtained by setting y=10-x.  These possibilities are:

 

(1,8) (2,7) (3,6) (4,5) (5,4) (6,3) (7,2) (8,1)

 

The product of these pairs is maximized by (4,5) and (5,4), each giving xy=20.

 

(a) is not true.

 

(b) is not necessarily true, e.g., (1,8)

 

(c) is not necessarily true, e.g., (1,8)

 

(d) is true because it corresponds to the line y=-x+5, which is parallel to the boundary lines and between them.

 

(e) is not true because it corresponds to the line y>x-8, which has a slope different from the boundary lines, hence crosses them and departs from the feasible region.