Graph the solution set of each system of inequalities. { -(4)/(3) x+y ≥-5 -4 x+3 y ≤13

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

Key Concepts

System of Inequalities

This concept involves working with a set of inequalities that are considered simultaneously. The goal is to determine the values that satisfy all of the inequalities in the system, which usually results in a region of feasible solutions on a graph. Each inequality imposes a constraint on the variable values, and the solution to the system is the intersection of all these constraints.

Graphing Linear Inequalities

Graphing linear inequalities involves drawing the boundary line represented by the corresponding equality and then shading the region that satisfies the inequality. It requires understanding how to transform an inequality into its boundary line and determine which side of the line represents the solution by using test points or analyzing the inequality's structure.

Boundary Lines

Boundary lines are the lines obtained by replacing the inequality sign with an equality. They serve as the border between different regions on the graph. The line is drawn either solid (if the inequality includes equality, such as ? or ?) or dashed (if the equality is not included, such as > or <), providing a clear demarcation of where the inequality changes from true to false.

Feasible Region

The feasible region is the area on the graph where all individual inequalities in the system overlap. This region represents all solutions that satisfy every inequality in the system. Finding the feasible region involves determining the intersection of the shaded half-planes resulting from each inequality.

Transcript

00:01 Here in this problem we have to graph the solution set for the given set of inequalities which are minus 4 divide by 3 times x plus y is greater than or equals to negative 5 and the second inequality is minus 4 x plus 3y is less than or equals to 13 and x plus y is less than or equals to 13 and x plus y is equal is greater than or equals to 2 now first of all i'll simplify the first inequality and i'm going to multiply the both sides of the inequality by 3 so we get negative 4x plus 3 y is greater than or equals to my minus 15 now i'll graph the solution set of this inequality and to do so i'll be graphing the equation minus 4x plus 3y is equals to 15 so first of all i'll make a table and by using that table we can graph this equation easily so this is this is a table now i'll be taking different value of x to obtain a different value of y so i'm taking x as 0 so we'll get y as minus 5 and now i'm taking a x as 3 .75 so i'm getting y as 0 you can take different value of x also you can take 1 2 3 it's up to you i've taken these value so i've obtained 2 points now i'll be plotting these two points on a graph paper so let's take a graph paper first so this is a graph paper now the first point is 0 comma negative 5 so 0 and negative 5 this point will represent 0.

02:14 Negative 5 and the next point is 3 .75 .0 so x is 3 .75 so almost this point will represent 3 .75 .0 i will join these two points to make a line.

02:34 It should be a straight line.

02:37 You can use your ruler also to draw this line.

02:42 Okay finally i've drawn a straight line almost.

02:50 Now the solution set of the first inequality would lie either this side or this side.

02:58 So we have to check but whether this side is the solution set for the given inequality or this side.

03:04 For that we have to take a point on either side of the line but not on the line.

03:09 So i'm taking the point on origin because it's very convenient to check on this point and whether the inequality fails or not fails at this point.

03:19 So the point is 0 .0 we know that the origin is 0 .0 so we're going to plug all these value here.

03:32 So x is 0 and y is also 0.

03:34 So we'll get 0 is greater than or equals to minus 5.

03:38 So the inequality is true here that 0 is greater than 5 is true.

03:46 So the solution set for this inequality would lie towards this side, towards the origin.

03:53 Because at this point in this region the inequality is true it doesn't fails now same now we'll take the second inequality which is negative 4x plus 3y is less than or equals to 13 now i'll be graphing the equation minus 4x plus 3y is equals to 13 and again i'll be using a table so let's take a table first so this is a table now we'll take different value of x to obtain different value of y so i'm taking x as 0 so we are getting y as 4 .33 13 divided by 3 is almost 4 .33 now i am taking the value of x as minus 3 .25 and we'll get the value of y as 0 now we're going to plot these value on the graph paper so the first point is 0 .4 .33.

05:02 So this is 0 and this point will represent 0 .4 .33.

05:09 Now the second point is minus 3 .25 .0...

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