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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.
What else can I help you with?
Can the solution of a system of linear inequalities be the points on a line?
yes
A system of two linear inequalities has either no points or infinitely many points in its solution?
the answer is true
which graph shows the solution to the system of linear inequalities y>2x+1 y?
3
What is the definition of equivalent inequalities?
The definition of equivalent inequalities: inequalities that have the same set of solutions
Inequalities containing two inequality symbols are called what?
Linear inequalities in one variable
When does A system of two linear inequalities have a solution?
When there is an ordered pair that satisfies both inequalities.
How are linear inequalities and linear equations the same?
They are not. An inequality cannot, by definition, be the same as an equation.
How many solution sets do systems of linear inequalities have Must solutions to systems of linear inequalities satisfy both inequalities In what case might they not?
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.
Can the solution of a system of linear inequalities be the points on a line?
yes
Is it possible for a system of two linear inequalities to ha a single point as a solution?
yes it is possible for a system of two linear inequalities to have a single point as a solution.
When is it possible for a system of two linear inequalities to have no solution?
A system of two linear inequalities can have no solution when the inequalities represent parallel lines that do not intersect. This occurs when the lines have the same slope but different y-intercepts. In such cases, there is no set of values that can satisfy both inequalities simultaneously, resulting in an empty solution set.
How many solution sets do systems of linear inequalities have. Must solutions to systems of linear inequalities satisfy both inequalities. In what case might they not?
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.
Do solutions to systems of linear inequalities need to satisfy linear inequalities?
No. For example, the solution to x ≤ 4 and x ≥ 4 is x = 4.
When is system of linear inequalities have no solution?
When the lines never intersect, usually when they are parallel.
What is the difference between the solution of a system of linear inequalities and the solution of a system of linear equations?
The solution of a system of linear equations consists of specific points where the equations intersect, typically yielding a unique point, infinitely many points, or no solution at all. In contrast, the solution of a system of linear inequalities represents a region in space, encompassing all points that satisfy the inequalities, often forming a polygonal shape in two dimensions. While equations define boundaries, inequalities define areas that can include multiple solutions. Thus, the nature of their solutions differs fundamentally: precise points versus expansive regions.
A system of two linear inequalities has either no points or infinitely many points in its solution?
the answer is true
When solving a system of linear inequalities what does the region that is never shaded represent?
It represents the solution set.
