How to Complete a Linear Inequality Assignment Without Full Instructions?

[wael_khayati]

hey so, this is an algebra assignment that we had to do and i really didn't understand the course material that well, but i managed to do the very first steps. anyways i was hoping you guys could help me finish the rest of this table. https://ufile.io/jowwrfj3 or you can see the file attached

 

[scottdave]

Please show what steps you have already taken, so we can try to guide you along.

 

[wael_khayati]

i first turned the inequalities into simple equations by adding the slack variables (u1 and u2) then constructed the table as seen above with the number i have respectively. the next part has something to do with pivoting or something but here's an example of another system you can follow

 

[PeroK]

@wael_khayati what is the question here? What are you trying to do with these inequalities?

 

[Mark44]

@wael_khayati what is the question here? What are you trying to do with these inequalities?

It appears to be a linear programming optimization problem, but without the objective function, and only the constraints shown.

 

[scottdave]

It's not much you can do with it, except tell what range of values are acceptable for x₁, x₂, and x₃.

If they said something like find a maximum, then that would be something to Solve.

 

[Mark44]

It's not much you can do with it, except tell what range of values are acceptable for x₁, x₂, and x₃.

If they said something like find a maximum, then that would be something to Solve.

After thinking about the problem for a bit, the solution would be the set of points in the first octant that satisfy the given inequalities.
My guess is that this exercise is the run-up to a linear programming problem, with the goal being to determine what will be called the feasible region, when the OP gets full-blown optimization problems. In this case, solving the problem is finding or sketching the region in space described by the inequalities.

 

[scottdave]

After thinking about the problem for a bit, the solution would be the set of points in the first octant that satisfy the given inequalities.
My guess is that this exercise is the run-up to a linear programming problem, with the goal being to determine what will be called the feasible region, when the OP gets full-blown optimization problems. In this case, solving the problem is finding or sketching the region in space described by the inequalities.

That makes some sense. I guess he is supposed to find a range for those "slack variables" which satisfy the "equation" that was made. For 2 dimensions it would be pretty easy to sketch out. 3 or higher... not so easy.

 

[Mark44]

That makes some sense. I guess he is supposed to find a range for those "slack variables" which satisfy the "equation" that was made. For 2 dimensions it would be pretty easy to sketch out. 3 or higher... not so easy.

I think the OP did a lot of unnecessary work by adding in the slack variables. My advice is to solve the system of inequalities exactly as it's given, by graphing the solution set. The first two inequalities define two half-planes that intersect in a line. The 3rd, 4th, and 5th inequalities constrain the solution set to the first octant.

 

[scottdave]

Those slack variables do look like busy work, but perhaps that's how the class is teaching this.
Can you expand on this @wael_khayati ? Thanks

 

[haruspex]

I think the OP did a lot of unnecessary work by adding in the slack variables. My advice is to solve the system of inequalities exactly as it's given, by graphing the solution set. The first two inequalities define two half-planes that intersect in a line. The 3rd, 4th, and 5th inequalities constrain the solution set to the first octant.

Isn't the use of slack variables standard in pivot table methods?
(But I don't understand how one can determine a pivot without an expression to be optimised.)

 

[Mark44]

Isn't the use of slack variables standard in pivot table methods?
(But I don't understand how one can determine a pivot without an expression to be optimised.)

Regarding slack variables, I believe that you are making the same mistake that the OP is making in this thread. In post #1, the instruction is "Solve this inequality."

As I said before, I believe the purpose of this problem is a precursor to linear programming problems, to give the student some practice in visualizing the feasible region. After the student has shown an understanding of the feasible region (i.e., the solution set for the inequalities), then an objective function can be introduced.

The problem as given here is simply to find the solution set for a set of inequalities, and has nothing to do with pivot tables. The OP is jumping the gun by adding slack variables and forming a tableau and otherwise going through the motions of solving a linear programming problem.

 

[haruspex]

Regarding slack variables, I believe that you are making the same mistake that the OP is making in this thread. In post #1, the instruction is "Solve this inequality."

As I said before, I believe the purpose of this problem is a precursor to linear programming problems, to give the student some practice in visualizing the feasible region. After the student has shown an understanding of the feasible region (i.e., the solution set for the inequalities), then an objective function can be introduced.

The problem as given here is simply to find the solution set for a set of inequalities, and has nothing to do with pivot tables. The OP is jumping the gun by adding slack variables and forming a tableau and otherwise going through the motions of solving a linear programming problem.

In post #3, the OP gives what appears to be an example provided by the teacher. There are slack variables and pivots, but no evidence of a target expression to be optimised nor any basis on which the pivots are chosen.
(And if I'm reading it correctly it ends with x1=x2=30, in violation of one of the given constraints. Maybe it's unfinished since one of the slack variables has a negative coefficient there.)

Whatever procedure is being followed in that example, the OP seems to believe, quite reasonably, that the same is to be applied to the problem in post #1.

 

[Mark44]

In post #3, the OP gives what appears to be an example provided by the teacher.

Not necessarily. I believe it's just another example that the OP found. According to post #1, the instruction is to "solve the inequality," which has nothing to do with slack variables or pivot tables.

There are slack variables and pivots, but no evidence of a target expression to be optimised nor any basis on which the pivots are chosen.

Since there is no objective function given, I believe the OP thinks he's supposed to work the problem as if it were a linear programming problem -- one in which an objective function is to be optimized in some way, given a set of constraints. All we have here is a system of inequalities, which makes me believe that what the problem says to do is exactly what should be done -- "solve the inequality (sic)."

Whatever procedure is being followed in that example, the OP seems to believe, quite reasonably, that the same is to be applied to the problem in post #1.

If we assume that both examples are linear programming problems, then in neither case has the OP given us the full problem. I was unable to open either the Word attachment or the link in post #1.

Let's hold off on further speculation until the OP comes back...