Charles earned an extra $20 for doing some extra errands around the house, and decided to spend some of it on almonds and banana chips, his favorite snacks.
The almonds sell for a unit price of $5 per pound. If Charles buys a pounds of almonds, the price he will pay is 5a dollars.
The banana chips are $3 per pound, so b pounds would have a cost of 3b dollars.
Charles total cost for his favorite snacks is 5a + 3b dollars. Since Charles has only $20 to spend, the inequality 5a + 3b ≤ 20 describes Charles’ situation in a mathematical form.
This inequality can be represented in a rectangular coordinate system by using x and y in place of a and b, respectively. The points on the line all represent combinations of almonds and chips that cost Charles exactly $20.
Points not on the line represent combinations of almonds and chips that cost either less than or more than $20. Any two points on the same side of the line will represent the same case of either being over or under $20, even though their difference from $20 may be different.
Take some time to look at the examples.
Once we know on which side of the line any one solution lies, we know on which side they all lie.
Now, Charles can buy fractions of a pound of almonds or fractions of a pound of banana chips, so not only are the points with whole number values for x and y also solutions, but so are the points with fractional values as well.
All of the solutions are indicated by shading the side of the line where the solutions are. Solutions on the boundary are indicated by drawing it with a solid line. In cases where the inequality is strict, the boundary is not part of the solution. This is indicated in the graph by drawing the boundary with a dashed line.
Charles may be able to buy 1.5 pounds of almonds or 2.25 pounds of banana chips, but he cannot buy a negative amount of either of these items, so there are extra conditions on the variables, the non-negativity constraints.
This restricts the solution to the first quadrant, since at least one variable is negative in each of the other quadrants. All of the possible solutions for Charles’ snack budget are contained in the shaded region.
This simulation allows you to experiment with systems of linear inequalities. You can graph up to four inequalities with the variables x and y.
Enter an inequality by typing the coefficients in the boxes or by using the sliders
You can change those values, and you can also drag the line in the graphing area.
Choose the type of inequality symbol you want to use from the drop-down control.
You can also select non-negativity constraints by using the checkbox,
This checkbox will shade the region where the individual shaded areas overlap.
The solution set is graphed.
Be sure to try the exercises and the questions. If a line gets out of control, check the values you are trying to use carefully. Good luck!