Graph the solution set of each system of inequalities. { -2 x+3 y ≤ 7 -2 x+9 y ≥

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

Key Concepts

Systems of Linear Inequalities

This concept involves solving multiple linear inequalities at once, where the solution is the set of points that satisfy every inequality simultaneously. It requires understanding how different inequalities interact and overlap, resulting in a region known as the feasible set or solution region.

Linear Inequalities in Two Variables

Linear inequalities in two variables involve expressions such as ax + by ? c or ax + by ? c. Each inequality represents a relationship between the variables and defines a region on the coordinate plane, indicating all the points that satisfy the inequality.

Graphing Linear Inequalities

Graphing linear inequalities means plotting the boundary line defined by the corresponding equality and then shading the region (half-plane) that represents the inequality. This visual approach helps in understanding and identifying the solution set of the inequality.

Boundary Lines and Half-Planes

The boundary line of a linear inequality is derived from converting the inequality to an equality. It partitions the plane into two half-planes. Determining which side of the boundary line to shade is essential, as it represents the set of points that satisfy the inequality.

Feasible Region

The feasible region is the intersection of all the half-planes defined by the system of inequalities. It represents all the points that simultaneously satisfy every inequality in the system. Graphically, it is the common area shared by the individual regions.

Transcript

00:01 Here in this problem we have to graph the solution set of the given set of inequalities which are minus 2x plus 3y is less than or equals to 7 and minus 2x plus 9y is greater than or equals to 1 and the last one is 2x plus 3y is less than or equals to 11 now, first of all, i'll graph the solution set for the first inequality.

00:39 And to do so, i'll be graphing the equation minus 2x plus 3y is equals to 7.

00:47 Now first of all, i'll make a table and by using that table, we can plot this equation or you could say the line on a graph paper easily.

00:58 So this is a table.

00:59 Now i'll take different values of x and obtain the different value of y so by using that points will able to make the graph easily okay when i'm taking x as zero so i'm getting y as 7 divided by 3 which is approximately 2 .33 and when i substitute x as minus 3 .5 here the value of y comes out to be as 0 now we have obtained these two points.

01:35 Now let's locate these point on the graph paper.

01:38 So i'll take a graph paper now.

01:41 So this is a graph paper.

01:43 Now the first equation is sorry, the first point is 0 .2 .3.

01:48 So here it is 0 and x is 0 and y is 2 .33.

01:52 So it will be here, something here.

01:56 This point will represent 0 .2 .33.

02:02 Now the next point is minus 3 .5 comma 0.

02:06 So minus 3 .5 would be this point and this point would be minus 3 .5 comma 0.

02:18 Now we have obtained two points.

02:19 Now let's join these two points to make a line.

02:23 It should be a straight line passing between these two points.

02:29 So this line represents the equation or you could see the line minus 2x plus 3y equals to 7 now the inequality region for the given first inequality would be this side or this side this side would be the solution or it can be this side also we have to check for that we have to consider any point on either side of the line this side or this side so i'm taking this side and i'm taking a point on the origin that is 0 .0 it is very convenient to check at this point so i'll substitute the values of x and y because we know that x coordinate is zero here y coordinate is also zero so i'll substituting here so we'll get 0 plus 0 is less than or equals to 7 which is true 0 is less than 7 that means the solution set for this inequality would be this side towards the origin because the inequality is true here now same thing we'll do for the second inequality so we'll draw we'll graph the equation minus 2x plus 9y is equals to 1 now again let's take a table so this is a table now when i substitute the value of x as 0 .5 here minus 0 .5 will get the value of y as 0 you can take different values i'm taking negative 0 .5 you can take 1 also 2 also it's up to you okay now next i'm taking x as 4 so i'll get y as one i get y as one so again we have obtained these two points now let's plot these point in a graph paper so the first point is negative point five and zero so this point would be would represent negative point five and zero and the second point is four comma one so here it is four and here it is one this point will represent four comma one now let's join these two points.

04:46 It should be a straight line.

04:49 You can use your ruler also.

04:51 But here i'm using my free hand...

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