Linear Programming

Step 1: Write the Constraints as Equations.
We start by converting each inequality into an equation in order to graph them: 1. 2. 3. 4. and are just the axes of the coordinate plane. Now we plot the lines corresponding to these equations.
Step 2: Find the Intersection Points.
To identify the feasible region, we solve for the intersection points of the lines by solving the corresponding systems of equations. Intersection of and :
Solving the system of equations: From the second equation, we get . Substitute this into the first equation: Substitute into : However, since , this intersection does not lie within the feasible region. Intersection of and :
Substitute into the first equation: Thus, the intersection point is . Intersection of and :
Substitute into : Thus, the intersection point is .
Step 3: Plot the Feasible Region.
Plot the points of intersection and , and draw the lines , , and . The feasible region is bounded by these lines.
Step 4: Evaluate the Objective Function at the Corner Points.
We now evaluate the objective function at the corner points of the feasible region, which are , , and . At : At : At :
Step 5: Conclusion.
Thus, the minimum value of is at , and the maximum value of is at .

This is a linear programming problem (LPP). The feasible region is bounded by the lines: 1. \Rightarrow 2. \Rightarrow 3. \Rightarrow

Step 1: Find corner points of the feasible region. - Intersection of and : Substitute in : - Intersection of and : - Intersection of and :

Step 2: Evaluate objective function at corner points. - At : - At : - At :

Step 3: Find minimum. The minimum value is

Final Answer:

Step 1: Simplify constraints.
So effective constraints are:

Step 2: Find corner points.
- When : \Rightarrow , and \Rightarrow . So feasible point . - When : \Rightarrow , and . So feasible point . - Intersection of and : From (2), . Substitute into (1): So intersection point .

Step 3: Evaluate objective function at feasible points.
- At : - At : - At : - At :

Step 4: Conclusion.
Maximum value of is

We have the system of inequalities:

Step 1: Convert inequalities into equations.

Step 2: Find intercepts. For (1): - If - If For (2): - If - If

Step 3: Plot lines and shade feasible region. - Line (1): passes through and . - Line (2): passes through and . - Feasible region lies in the first quadrant bounded by these lines and coordinate axes.

Step 4: Find corner points of feasible region. By solving (1) and (2): Multiply first by 3: Multiply second by 4: Subtract: Substitute in (1): So, point of intersection is .

Step 5: Vertices of feasible region. Conclusion: The shaded feasible region is the quadrilateral formed by these points in the first quadrant.

Final Answer: Feasible region vertices are

We are asked to maximize the objective function subject to the following constraints:

Step 1: Plot the constraints.
The constraint represents a region below the line . The inequalities and restrict the solution to the first quadrant. The feasible region is a triangle with vertices at , , and .

Step 2: Evaluate at the vertices.
We now evaluate at the three vertices of the feasible region: - At , . - At , . - At , .

Step 3: Find the maximum value.
The maximum value of occurs at , where . Conclusion: The maximum value of is , which occurs at the point .

We need to maximize , subject to the constraints: We begin by graphing the constraints: 1. represents a line with slope 1 and intercept -1. 2. represents a line with slope 1 and intercept 0. 3. is the region to the right of the y-axis. 4. is the region above the x-axis. The feasible region is the area bounded by these lines. We can find the vertices of this region by solving the system of equations: 1. Solve and : From (ii), . Substituting into (i): which is a contradiction, meaning there is no solution. Next, solve for other pairs of constraints to determine the vertices.

Step 1: Graph the inequalities.
To solve the LPP graphically, plot the following lines: and the boundaries and . Find the intercepts for each line: - For Line 1, when , , so . When , . - For Line 2, when , , so . When , . Plot the lines on a graph.

Step 2: Identify the feasible region.
The feasible region is the area where all constraints overlap.

Step 3: Evaluate the objective function at the corner points.
From the graph, the corner points are identified, and the objective function is evaluated at each corner.

Step 4: Find the minimum value of .
By evaluating at each corner point, we find the minimum value of . Conclusion: The minimum value of is found at one of the corner points of the feasible region.