Question

Show that the solution set of the following system of linear inequalities is an unbounded region ` 2x+y ge 8, x+2y ge 10, x ge 0, y ge 0`.

Answer 1

Consider the inequation ` 2x+y ge 8` as an equation, we have
` 2x+y = 8`
` rArr y=8 -2x`
`{:(x,0,4,3),(y,8,0,2):}`
The line `2x+y=8` intersects coordinate ases at (4, 0) and (0, 8).
Now, point (0, 0) does not satisfy the inequation ` 2x+y ge 8`.
Therefore, half plane does not contain origin.
Consider the inequation ` x+ 2y ge 10` , as an equation, we have
`x+2y=10`
` rArr 2y =10-x`
`{:(x,10,0,8),(y,0,5,1):}`
The line `2x +y=8` intersects the coordinate axes at (10, 0) and (0, 5).
Now, point (0, 0) does not satisfy the inequation ` x+2y ge 10`.
Therefore, half plane does not contain (0, 0).
Consider the inequation `x ge 0 and y ge 0 ` clearly, it prepresents the region in first quadrant.
The graph of the above inequations is given below

It is clear from the graph that common shaded portion is unbounded.