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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.

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Q: What is the definition of the solution of linear inequalities?

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yes

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The definition of equivalent inequalities: inequalities that have the same set of solutions

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Linear inequalities in one variable

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When there is an ordered pair that satisfies both inequalities.

yes it is possible for a system of two linear inequalities to have a single point as a solution.

They are not. An inequality cannot, by definition, be the same as an equation.

yes

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.

There is only one solution set. Depending on the inequalities, the set can be empty, have a finite number of solutions, or have an infinite number of solutions. In all cases, there is only one solution set.

No. For example, the solution to x â‰¤ 4 and x â‰¥ 4 is x = 4.

When the lines never intersect, usually when they are parallel.

Inequalities tend to have infinitely many solutions.

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It represents the solution set.

A system of linear inequalities

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