Determine graphically the solution set for each system of inequalities and indicate whether the

step 1 All righty, four to six. Got those colored pencils already. So we can have some kind of fun. So

Determine graphically the solution set for each system of...

Key Concepts

Graphical Representation of Constraints

Graphically representing constraints means plotting each individual inequality on a coordinate system, where each constraint divides the plane into regions. The solution to the system of inequalities is found by identifying the area where all these individual regions intersect.

Intersection of Half-Planes

Each linear inequality represents a half-plane. The intersection of these half-planes constitutes the feasible region that satisfies all the inequalities simultaneously. This concept is crucial in understanding systems of inequalities in geometry and optimization problems.

Feasible Region

The feasible region is the set of all points that simultaneously satisfy all constraints in a system of inequalities. It can be visualized as a shaded area on a graph where all the half-plane conditions overlap and is central to solving problems in linear programming and systems of equations.

Bounded and Unbounded Regions

This concept differentiates between regions that are completely enclosed by boundaries (bounded) and those that extend infinitely in at least one direction (unbounded). Determining whether a feasible region is bounded or unbounded is an important aspect of graphical solutions and optimization analysis.

Inequalities in Two Dimensions

This concept involves understanding and representing inequalities involving two variables graphically on the coordinate plane. It includes recognizing that an inequality like x + y ? 6 defines a half-plane rather than a single line, indicating all the points that satisfy the condition.

Transcript

00:02 All righty, four to six.

00:08 Got those colored pencils already.

00:11 So we can have some kind of fun.

00:14 So i'm going to start with the first boundary, which is x is greater than equal to six.

00:22 Seeing that before somewhere, i haven't checked.

00:25 Which is still the vertical line, going down the y -axis.

00:31 And we are still going to the right.

00:35 Now your second one, well, look you there.

00:39 It's why is greater than or equal to zero.

00:42 That should have been a six up there.

00:44 Why the heck did i put a, let me should be a zero? why did i put a six? and why is that red? you're probably thinking, this woman is nuts.

00:56 Well, you know, that's what happens when you do 50 of these.

01:00 Hopefully you're not having to do all 50 of them.

01:03 It'd be overkill.

01:05 And then we're going to then again.

01:07 Maybe, you know, practice, mix.

01:08 Perfect i'm sure basketball players shoot more than 50 free throws when they want to get really good i don't know so 3x that still don't know what that starts out is right 3x plus y is less than equal to 6 so if x is 0 y is 6 if y is 0 then x is 2 so 0 6 1 2 so 0 6 1 2 three, four, five, six, and two, zero.

01:53 So let's draw that little line.

02:10 And we have zero less than equal six, so that's true.

02:17 So it looks like this is gonna be shading that direction.

02:22 I'm gonna move this over here, because i need it to be in here...

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