Graph the solution set of each system of inequalities. { x+y ≤ 4 -x+y ≤ 4 x+5 y

Graph the solution set of each system of inequalities. { ...

Key Concepts

Intersection of Regions

The intersection of regions refers to the overlapping area where all individual conditions or inequalities are met simultaneously. This common 'intersection' is critical in solving systems of inequalities, as it defines the set of points that satisfy the entire system.

Feasible Region

The feasible region is the common area that satisfies all the inequalities in a system. It is found by overlapping the shaded regions corresponding to each inequality. This region represents all possible solutions to the system of inequalities.

Half-Planes

A half-plane is one of the two regions into which a boundary line divides the plane. For a linear inequality, one of these regions corresponds to the solution set, where every point satisfies the inequality. Identifying the correct half-plane is a key step in graphing inequalities.

Graphing Inequalities

Graphing inequalities involves drawing lines or curves representing the equality part of the inequality, and then shading the region where the inequality holds true. This visual representation helps in understanding and identifying the set of solutions to the inequality.

System of Inequalities

A system of inequalities consists of multiple inequalities that must be satisfied simultaneously. Each inequality establishes a condition on the values of the variables, and the solution to the system is the set of all points that meet every individual inequality in the system.

Boundary Lines

Boundary lines are the lines drawn for the equality part of each inequality. They indicate the transition between regions that satisfy and do not satisfy the inequality. Whether the line itself is included in the solution depends on whether the inequality is strict (using < or >) or inclusive (using ? or ?).

Transcript

00:01 Here in this problem we have to graph the solution set of the given set of inequalities which are x plus y is less than or equals to 4 negative x plus y is less than or equals to 4 x plus 5 y is greater than or equals to 8 so we have to graph these inequalities the solution set of these inequality on this inequality on graph paper so for that i'll be graphing the solution set for the first inequality and to do so i'll be graphing the equation x plus y is equals to 4 now i'll make a table so that by using that table we can graph this equation easily so this is a table now i'll take different values of x and we'll obtain different value of y for that and you can take your own values also i'm taking x as 0 you can take 1 2 3 4 it's up to you so when i take x as 0 i'll get y as 4 now when i get when i take x as 4 i get y as 0 now we have obtained these two points now we're going to plot on a graph paper so now i'll take a graph paper so this is a graph paper now first point is 0 .4 so x coordinate is 0 and y coordinate is 4 so this point will represent 0 .4 now second is 4 .0 so this point will represent 4 .0 now we'll join both the points to make a line and it should be a straight line you can use your ruler also to draw this line actually it should be fine i get it now same we have obtained the equation of the line for this equation now the solution set for our given inequality the first inequality would lie either on this side or this side it depends so we have to check it whether this side is the solution set for our inequality or this side and to do so we have to take a point and it should be on either side of the line this side also this side also but not on the line so i'm taking origin as that point and it is very convenient to take this point because here x and y coordinate is zero now we're going to plug the value of x and y coordinate here in the inequality so zero plus zero which is zero zero is less than or equals to four this inequality is true here is less than four it is true so the solution set for this inequality would be in this side towards the origin now we'll solve for second inequality and we'll perform the same thing.

03:32 So i'm taking i'll grab the equation minus x plus y is equals to 4.

03:38 Now i'll take a table.

03:42 So this is a table.

03:44 Now we'll take different value of x and we'll get different value of y for that.

03:49 Okay, now i take x as zero here.

03:53 So i get y as 4.

03:55 Now if i take x as minus 4.

04:00 So i get y as 0.

04:04 And i will plot this point in the graph paper.

04:07 So first point is 0 .4.

04:10 0 .4 is this point...

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