Study guides

☆☆

Q: How do you know which region of the graph of a system of linear inequalities contains the solutions?

Write your answer...

Submit

Still have questions?

Related questions

A solution to a linear inequality in two variables is an ordered pair (x, y) that makes the inequality a true statement. The solution set is the set of all solutions to the inequality. The solution set to an inequality in two variables is typically a region in the xy-plane, which means that there are infinitely many solutions. Sometimes a solution set must satisfy two inequalities in a system of linear inequalities in two variables. If it does not satisfy both inequalities then it is not a solution.

Systems of inequalities in n variables with create an n-dimensional shape in n-dimensional space which is called the feasible region. Any point inside this region will be a solution to the system of inequalities; any point outside it will not. If all the inequalities are linear then the shape will be a convex polyhedron in n-space. If any are non-linear inequalities then the solution-space will be a complicated shape. As with a system of equations, with continuous variables, there need not be any solution but there can be one or infinitely many.

Each linear equation is a line that divides the coordinate plane into three regions: one "above" the line, one "below" and the line itself. For a linear inequality, the corresponding equality divides the plane into two, with the line itself belonging to one or the other region depending on the nature of the inequality. A system of linear inequalities may define a polygonal region (a simplex) that satisfies ALL the inequalities. This area, if it exists, is called the feasible region and comprises all possible solutions of the linear inequalities. In linear programming, there will be an objective function which will restrict the feasible region to a vertex or an edge of simplex. There may also be a further constraint - integer programming - where the solution must comprise integers. In this case, the feasible region will comprise all the integer grid-ponits with the simplex.

Linear programming is just graphing a bunch of linear inequalities. Remember that when you graph inequalities, you need to shade the "good" region - pick a point that is not on the line, put it in the inequality, and the it the point makes the inequality true (like 0

It represents the solution set.

An inequality determines a region of space in which the solutions for that particular inequality. For a system of inequalities, these regions may overlap. The solution set is any point in the overlap. If the regions do not overlap then there is no solution to the system.

Yes. If the feasible region has a [constraint] line that is parallel to the objective function.

A linear equation corresponds to a line, and a linear inequality corresponds to a region bounded by a line. Consider the equation y = x-5. This could be graphed as a line going through (0,-5), (1,-4), (2,-3), and so on. The inequality y > x-5 would be the region above that line.

linear in active region....

the feasible region is where two or more inequalities are shaded in the same place

In 2-dimensional space, an equality could be represented by a line. A set of equalities would be represented by a set of lines. If these lines intersected at a single point, that point would be the solution to the set of equations. With inequalities, instead of a line you get a region - one side of the line representing the corresponding equality (or the other). The line, itself, may be included or excluded. Each inequality can be represented by a region and, if these regions overlap, any point within that sub-region is a solution to the system of inequalities.

A feasible region is, in a constrained optimization problem, the set of solutions satisfying all equalities and/or inequalities. On the other hand a linear programming is a constrained optimization problem in which both the objective function and the constraints are linear, therefore a feasible region on a linear programming problem is the set of solutions of the a linear problem. Many algorithms had been designed to successfully attain feasibility at the same time as resolving the problem, e.g. reaching its minimum. Perhaps one of the most famous and extensively utilized is the Simplex Method who travels from one extremal point to another, which happens to be the possible extrema given the convex nature of the problem, by maintaining a fixed number of components to zero, called basic variables. Then, the algorithm arrives to a global minimum generally in polinomial time even if its worst possible case has already been proved to be exponencial, see Klee-Minty's cube.

People also asked