Question

Describe the boundary lines for two- variable inequalities. Why are the boundary lines for two- variable inequalities with greater than and less represented by dotted lines? Provide examples.
First, define a boundary line and tell where it comes from. Then, describe what the boundary line can tell us about solutions to an inequality. You can also talk about how to know what part of a graph to shade. Finally , talk about the cases where we use each type of boundary line. ( solid and dotted/ dashed).
Real - life Relationship: If you have 100 $ available to buy party favors ( 3$ per bunch of balloons and 4 $ per bag of candy) than you can solve an inequality to find the possibilities . If x = # of bunches of balloons and y= number of bags of candy then we want to solve: 3x+ 4y<=100.
Some possible solutions are: no bunches of balloons and 25 bags of candy,20 bunches of balloons and 10 bags of candy. There are other possibilities!
Challenge: Imagine we have two boundary lines: one solid and one dashed. If they are not parallel is the point where they meet included in the solution? Why or why not?
If you are not sure, try an example, such as y < x + 1 and y < = 2x-4. Graph both boundary lines and find the point of intersection. Then , see if the coordinates satisfy both inequalities.

Answer 1

Two variables inequalities can be of the following types :

Type 1 . ax + by c or, ax + by c

Type 2 . ax + by    or, ax + by c where a,b,c are real numbers

Boundary line is a line that distinguishes the solution region from the non solution region. Boundary lines comes into play whenever a constraint is introduced.

Boundary lines can tell us about the type of ineqality , which is given below as,

Boundary lines for the inequalities of the Type 1 are SOLID LINES , while boundary lines for Type 2 inequalities are DOTTED/DASHED LINES.

Boundary lines for less than or greater than inequlaities are dotted lines because the points lying on the boundary line are excluded from the solution of the corresponding inequality.

To know what part of the graph to shade , we use the point (0,0) If the origin satisfies the inequality then the part of the graph which contains the origin has to be shaded , otherwise the part not containing the origin should be shaded.

EXAMPLE : 2x + 3y > 4 .

Solving 3x + 4y 100, from the graph given below , we can see that the solution of the given inequality is represented by the part of the graph lying in the first quadrant, so all the points in the red coloured region in the first quadrant represents the solution of the given ineqality.

Whenever a solid line and a dashed line interesect then the point of intersection is not included in the solution . The reasoning behind is this , points on the solid line satisfy the concerned inequality , but the points of the dotted line does not form part of the solution of inequality . Therefore it becomes very natural to understand that for system of two inequality , one represented by a solid line and other by the dotted line , the points of intersection does not form part of the solution.

Example : y< x+ 1 and y<= 2x-4 , below is the graph of these inequalities, blue graph represents y<= 2x-4 and red graph represents y<x+1

point of intersection is (5,6) , we can clearly see that (5,6) does not satisfy y<x+1