Many business and career problems can be represented using systems of inequalities. Let’s take a farmer who has a certain amount of time to plant his crops. He is planting tomatoes and beans, but in order to harvest them in time to meet the contract he has made with a canning company, he must plant within 21 days. He can plant beans at a rate of 200 acres a day, and he can plant tomatoes at a rate of 150 acres a day. Altogether, he will use, at most 2,000 acres. How many acres of each type of crop should he plant?
You could represent his problem in this way:
Let T represent the number of days planting tomatoes.
Let B represent the number of days planting beans.
You know that both T and B must be more than zero.
T + B ≤ 21
200B + 150T ≤ 2000
Now that you have seen how a problem can be represented, let’s talk about graphing systems. You will come back to solving the problem in the next part of this section.
Example 1: Graphing a System of Inequalities and Locating Solutions
Graph the system and name three possible solutions.
y < -2x + 4
y ≥ x – 1
Inequality
Boundary Line
y-intercept
slope
y < -2x + 4
broken
4
-2/1
y ≥ x – 1
solid
-1
1/1
Graph the first inequality’s boundary and shade below (because of <). Note that the boundary is a broken line. (For simplicity, you will not see the broken lines on these graphs.)
On the same graph, graph the second inequality’s boundary (solid line), and shade above (because of >).
The solutions are in the dark green area of the graph, where the light green and gray overlap.
Three possible solutions are any three ordered pairs in the darker green area, such as (-1, -1), (-2, 3), or (1, 1), and any points on the solid line in the dark green portion, such as (1, 0) or (-2, -3).
Example 2: Parallel Lines
Graph each of the following systems and name three ordered pair solutions.
2x + y > 3
2x + y < -3
The first graph shades above the broken line and the second shades below. As you can see, they do not intercept (overlap) at all.
No Solution
2x + y < 3
2x + y > -3
Although this appears to be the same system, the inequality signs were changed, so the direction of the shading changed. You can see that they intercept (overlap) in the orange area. Any ordered pair in the orange area is a solution to the system.
Three ordered pair solutions would be (-1, 3), (0, 0), and (2, -2).
Example 3: Horizontal and Vertical Lines
Graph the system and name three ordered pair solutions.
y ≤ 0
x ≥ -1
You can see that they overlap in the brownish area.
Three solutions are (-1, -2), (1, -3), and (2, 0).
Quick Practice
Solve each system by graphing. Give three solutions.
x > -4
y ≤ 1
You can name any three ordered pairs in the blue-green overlap area and the solid line portion contained in the area.
x ≥ 3
x + y ≥ 2
You can name any three ordered pairs in the dark pink area and the solid line portion contained in the area.
x – y ≥ 1
2x + y ≤ 5
Solve each in terms of y first.
y ≤ x – 1
y ≤ 5 – 2x
Solutions are in the yellow-green area and the two solid line portions contained in the area.
3x + y > -2
-3x + 2y < 1
Solve both for y first.
y > -3x – 2
y < 3/2x + 1
Solutions are in the dark-green area and the solid line portion contained in the area.
