An objective function and a system of linear inequalities representing constraints are given. Graph the system of inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region. Use these values to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Objective Function: z = 19x + 4y
Constraints: 0 < than or equal to x < than or equal to 10
0 < than or equal to y < than or equal to 5
3x + 2y greater than or equal to 6

To graph the system of linear inequalities representing the constraints, follow these steps:

1. Graph the lines representing the equations of the inequalities. Start by graphing the lines for the equations 0 ≤ x ≤ 10 and 0 ≤ y ≤ 5. These lines represent the boundaries for x and y, respectively.

- For 0 ≤ x ≤ 10, draw a vertical line passing through x = 0 and x = 10.
- For 0 ≤ y ≤ 5, draw a horizontal line passing through y = 0 and y = 5.
- Shade the region between these lines to represent the feasible values for x and y.

2. Graph the line representing the inequality 3x + 2y ≥ 6. To do this, convert the inequality to the equation 3x + 2y = 6 and plot the line.

- To find where the line intersects the x-axis, set y = 0 and solve for x: 3x + 2(0) = 6, x = 2.
- To find where the line intersects the y-axis, set x = 0 and solve for y: 3(0) + 2y = 6, y = 3.
- Plot these points (2, 0) and (0, 3), and draw a line passing through them.

3. Determine which side of the line to shade by picking a test point not on the line. For example, use the point (0, 0) and substitute its x and y values into the inequality:

- 3(0) + 2(0) ≥ 6? This statement is false.
- Since the inequality is false, shade the side of the line that does not contain the origin.

Now, you should have a graph representing the system of inequalities.

To find the values of the objective function at each corner of the feasible region, identify the points where the lines intersect:

1. Corners of the region:
- (0, 0)
- (0, 5)
- (2, 3)
- (10, 0)
- (10, 5)

2. Calculate the value of the objective function at each corner by substituting the x and y values:
- z = 19(0) + 4(0) = 0
- z = 19(0) + 4(5) = 20
- z = 19(2) + 4(3) = 50
- z = 19(10) + 4(0) = 190
- z = 19(10) + 4(5) = 230

Finally, determine the maximum value of the objective function and the corresponding values of x and y:

- The maximum value of the objective function is 230, which occurs at the point (10, 5).