Systems of linear inequalities are used all over, from animal conservation efforts to trying to decided how many gifts to buy.

Suppose you want to decide how many gifts you will buy for family and friends.

You can use a system of linear inequalities to help you decide on a good solution.

Suppose you want to buy between 3 and 5 gifts for your 2 family members and 3 friends.

You can represent this problem as two inequalities, shown below.

You can graph the inequalities to find solutions for how many different ways you can buy presents.

y +

x +

1

2

3

4

5

1 2 3 4 5-5

-4

-3

-2

-1 -

1

-2

The region where there is some overlap is where the solutions of the system of inequalities lie.

y +

x +

1

2

3

4

5

1 2 3 4 5-5

-4

-3

-2

-1 -

1

-2

Only the solutions which lie in the first quadrant can be used to decide how many gifts to by, since you cannot give a negative gift.

Only solutions having whole numbers are useful solutions, since you cannot give a partial gift.

y +

x +

1

2

3

4

5

1 2 3 4 5-5

-4

-3

-2

-1 -

1

-2

Red dots represent the actual solutions.

Buying just for family members

Buying for family and friends

Buying just for friends

You could by 1 present for each of your friends and none for your family.

You could by 2 presents for each of your family members and none for your friends.

You could buy 1 present for each of your friends and family members.