solve the following system of inequalities graphically
3x+2y<150, x+4y<80, x<80,x>0,y>0
draw each line. Then shade the area below the two slanting lines, above the x-axis, and between the y-axis and the vertical line at x=80.
The solution is the part where all the shaded areas intersect.
A good start is shown here:
http://www.wolframalpha.com/input/?i=solve+3x%2B2y%3C150%2C+x%2B4y%3C80
To solve the system of inequalities graphically, we need to plot the graphical representations of each inequality and observe the overlapping regions.
Let's start by graphing the first inequality: 3x + 2y < 150.
Step 1: Convert the inequality into an equation.
3x + 2y = 150.
Step 2: Rewrite the equation in terms of y.
2y = 150 - 3x.
y = (150 - 3x) / 2.
Step 3: Choose arbitrary values for x and solve for their corresponding y-values to obtain two points that lie on the line.
We can choose x = 0 and x = 50, for example.
For x = 0, y = (150 - 3(0)) / 2 = 75.
For x = 50, y = (150 - 3(50)) / 2 = 25.
Step 4: Plot the line using the obtained points (0, 75) and (50, 25). Draw a dashed line, as the inequality is strict (<), indicating that the points on the line are not included in the solution set.
Now, let's graph the second inequality: x + 4y < 80.
Step 1: Convert the inequality into an equation.
x + 4y = 80.
Step 2: Rewrite the equation in terms of y.
4y = 80 - x.
y = (80 - x) / 4.
Step 3: Choose arbitrary values for x and solve for their corresponding y-values to obtain two points that lie on the line.
We can choose x = 0 and x = 80, for example.
For x = 0, y = (80 - 0) / 4 = 20.
For x = 80, y = (80 - 80) / 4 = 0.
Step 4: Plot the line using the obtained points (0, 20) and (80, 0). Draw a dashed line to indicate that the points on the line are not included.
Next, let's graph the inequality x < 80.
Step 1: Draw a vertical line at x = 80. Make sure to draw a dashed line since the inequality is strict (<).
Finally, let's graph the inequality y > 0.
Step 1: Draw a horizontal line at y = 0. Make sure to draw the line solid, since the inequality is not strict (>).
Now, let's observe the overlapping region among all the inequalities. The solution to the system of inequalities will be within this region.
Based on the graph, it can be observed that the overlapping region formed by all the inequalities lies in the first quadrant, below the line 3x + 2y = 150, and to the left of x = 80.
Thus, the graphical solution to the system of inequalities is the region below the line 3x + 2y = 150, to the left of x = 80, and within the first quadrant.