Systems of linear inequalities

Example 1

Graph the solution to the system of inequalities

( i )

( ii )

Solution .

These inequalities are equivalent to

( i )

( ii )

The solution to ( i ) is the set of all points on or below the line which is shown in blue on the graph below.

The solution to ( ii ) is the set of all points on or above the line which is drawn in red on the graph below.

After shading in those points which are on or below the blue line and on or above the red line, we get the following graph. Note the solution set has been shaded yellow.

Note that the solution does not stop at x = -15. If we zoom out, we will see that it continues.

The solution in example one is unbounded .

Example 2

Graph the solution to the system

( i )

( ii )

( iii )

( iv )

Solution .

Note that inequalities (iii) and (iv) have been added on to the system used in example one. We must then intersect the region obtained in example one with the region containing non-negative x and y values. This gives the following solution.

The solution in example two is bounded .

Example 3

Graph the solution to the following system of inequalities.

( i )

( ii )

( iii )

( iv )

( v )

Solution .

The last two inequalities insure that we have non-negative x and y values. As a result our graph will be in the first quadrant. Solving the other three inequalities for y gives us

( i ) ( needs to be on or under the blue line )

( ii ) ( needs to be under the red line )

( iii ) ( needs to be on or above the green line )

Graphing these lines with the colors noted above, we get

Graphing the intersection of the five inequalities we get

Example 4

Graph the solution to the following system of inequalities.

( i )

( ii )

( iii )

( iv )

( v )

( vi )

( vii )

Solution .

In equalities ( i ) through ( v ) are the same as the ones used in example three. The addition of the restrictions that are given in ( vi ) and ( vii ) are reflected in the graph below.