A doctor needs at least 60 adults for a medical study. He cannot use more than 40 men in the study. Write a system of inequalities to model the situation and solve the system by graphing.
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Let's let x represent the number of men and y represent the number of women in the study.
Based on the given information, we can write the following inequalities:
1. x + y ≥ 60 (There need to be at least 60 adults in the study)
2. x ≤ 40 (There cannot be more than 40 men in the study)
To solve the system by graphing, we can first graph the line x + y = 60 and shade the region above or on the line to satisfy the first inequality. Then, we can draw a vertical line at x = 40 and shade the region to the left of the line to satisfy the second inequality.
The solution will be the area where the two shaded regions overlap.
Here is a graph to represent the system of inequalities:
```
y | ________________
| | |
60 | | |
| | |
| |X |
| | X |
| | X |
| | X |
| | X |
|__________________|_____________________
0 40 60 x
```
The overlapping region represents the valid solutions, where the number of men (x) is less than or equal to 40 and the total number of adults (x + y) is greater than or equal to 60.
To model the given situation, we need to establish two inequalities based on the given constraints.
Let's assume 'x' represents the number of men in the study, and 'y' represents the number of women in the study.
Inequality 1: The doctor needs at least 60 adults for the study:
x + y ≥ 60
Inequality 2: The doctor cannot use more than 40 men in the study:
x ≤ 40
To solve the system by graphing, we can start by rewriting both inequalities in slope-intercept form (y = mx + b).
Inequality 1:
x + y ≥ 60
y ≥ -x + 60
Inequality 2:
x ≤ 40
Now let's plot the graphs on a coordinate plane:
For the first inequality, y ≥ -x + 60, we start by graphing the line y = -x + 60. This line has a y-intercept of 60 (when x = 0) and a slope of -1 (since the coefficient of x is -1).
For the second inequality, x ≤ 40, we have a vertical line that intersects the x-axis at x = 40.
Now let's plot these lines on a graph and shade the appropriate region:
|
________|_______
| /
________|_____/
| /
________|___/
| /
________|_/
|
|
|
|
|
__________________|_______________________
|
0 40
Based on the graph, the shaded region above the line y = -x + 60 and to the left of x = 40 represents the feasible solutions that satisfy both inequalities. The doctor can select any point within this region.
Please note that the graph is an approximation and doesn't include all possible integer solutions. Additionally, since the doctor wants to select an integer number of men and women, we need to find the integer coordinates within the shaded region.
Therefore, to find an appropriate solution, we need to find the points (x, y) that fulfill the following conditions:
1. y is an integer.
2. x is an integer.
3. x ≤ 40.
4. x + y ≥ 60.
By examining the graph or systematically checking the integer coordinates within the shaded region, we can find the feasible solutions that satisfy the given conditions.