In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system

In Exercises 33-46, sketch the graph (and label the...

Key Concepts

Labeling Vertices

Vertices are important points where the boundary curves or lines intersect, forming the 'corners' of the solution region. Identifying and marking these vertices is crucial for precisely defining the shape and limits of the solution set.

Graphing Linear Inequalities

Graphing a linear inequality involves drawing its boundary line—in this case, x = y + 2—and determining which side of the line satisfies the inequality (x < y + 2). A common method is to choose a test point not on the line to see if it satisfies the inequality.

Finding Intersection Regions

This concept is concerned with obtaining the common area that satisfies all inequalities in a system. After the individual regions for each inequality are identified, their intersection represents the feasible solution area where all conditions hold simultaneously.

Graphing Systems of Inequalities

This concept involves understanding that a system of inequalities defines several regions in the coordinate plane and that the overall solution set is the intersection of those regions. To find the solution, each inequality is graphed separately, and the area where all conditions overlap is identified.

Graphing Parabolic Inequalities

When dealing with a parabolic inequality, the boundary is a parabola—here given by an equation like x = y². The inequality (for example, x > y²) specifies which side of the parabola is part of the solution set. A test point can be used to determine if the region to the left or right of the parabola should be shaded.

Transcript

00:01 So we are to sketch the system of inequalities and find any boundary points, points of intersection between the two.

00:08 Okay, so for the first one, you might not immediately recognize x greater than y squared.

00:17 Of course, the boundary curve is going to be x equals y squared.

00:21 You might not recognize that immediately, but i bet that you would recognize it if i replaced x and y.

00:26 So this would be y equals x squared.

00:29 Okay, well, that's obviously a parabola.

00:31 And there are two ways to go about it.

00:33 I'm just going to go and do the first way.

00:35 The first way is to recognize that this is simply the same thing as the parabola, which i'm actually going to sketch in green.

00:45 This would be the same thing as this parabola right here, reflected about the line y equals x.

01:01 So everything gets reflected over this line.

01:06 Okay? the other way to look at it is it's essentially taking this parabola and it's going to be rotating it by 90 degrees this way, clockwise.

01:19 It's the same thing.

01:21 And if you do that, if you can just kind of tilt this up a lot, you probably can't tilt your screens.

01:25 But if you draw this and just completely flip it 90 degrees, that's actually just going to be the same as a graph that looks like this.

01:34 So that's the graph of x equals y squared.

01:38 And in general, you can see that this is a complete reflection.

01:41 This graph is a complete reflection of this.

01:44 About this line y equals x.

01:48 So this is the only graph that we're concerned with, so just kind of ignore everything else i drew.

01:53 I know it's going to be a little hard to.

01:55 I can try to erase what i can.

01:58 But this still isn't our boundary curve, because the boundary curve is x is greater than and not equal to y squared.

02:05 So this actually has to be dashed and not a solid curve.

02:11 We have to figure out where to shade.

02:13 So since this is x is greater than y squared, i'm going to go and use a test.

02:17 Point, we can't use the origin as most people do because the origin actually is on this curve.

02:23 So i'm going to use something like 1 comma 0, which is right here.

02:28 So i'm going to plug in 1 comma 0 into my inequality right here and either it fails or it passes as the inequality.

02:39 Okay, if the inequality is true, that means i shade everything inside the parabola.

02:42 If it's false, i shade everything outside.

02:45 Okay, so if i replace x with 1 and y with 0, i get the statement 1 is greater than 0, which is true.

02:53 So actually i'm going to shade everything inside this region, not including the boundary curve itself.

03:01 Okay, and this is a line.

03:03 You can graph it pretty much the same way.

03:05 You can replace x and y and then flip it, or you can just solve for y, which in this case i think is much easier.

03:11 So in this case, we get that y is greater than, right? the greater than is pointing towards the y, and then i subtract two from both sides to get y is greater than x minus 2.

03:21 So negative 2 is going to be down here, and then x is going to be, it's going to look like that for my line.

03:38 Now, of course, this is a dashed and not a solid line because we're not including it...

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