Are there any points whose coordinates satisfy both y>3 x+1 and y<3 x-4 ? If so, give the coordi

Are there any points whose coordinates satisfy both y>3 x+1...

Key Concepts

System of Linear Inequalities

This concept refers to a collection of inequalities that involve linear expressions. When solving these systems, one seeks the set of points that satisfy all inequalities simultaneously, which typically corresponds to the overlapping region of multiple half-planes on a graph.

Feasible Region and Inconsistency

The feasible region is the set of all points that meet every constraint in a system of inequalities. In some cases, however, the given inequalities may be mutually exclusive or contradictory, resulting in an empty feasible region. This occurs when the conditions imposed by the inequalities cannot be satisfied at the same time, rendering the system inconsistent.

Graphical Representation of Inequalities

Graphical interpretation is an important tool in understanding systems of inequalities. Each inequality represents a half-plane defined by a straight line (its boundary). Visualizing these half-planes on a coordinate plane helps identify whether there is any overlapping region that satisfies all conditions simultaneously, or if the constraints contradict each other, leading to no possible solutions.

Transcript

00:01 If i want to know whether a coordinate can be a solution from two different inequalities, i could do substitution, just like with a system of equations, or i can graph it.

00:14 And i happen to find graphing to be a bit easier, just because it's a little more visual, and i think it takes a lot less time.

00:20 So i'm going to solve this question by graphing.

00:24 So my first inequality here, which is y is greater than 3x plus 1, i'm going to do in blue.

00:29 And so my b here, my y intercept is one.

00:35 So i'm going to graph that here.

00:37 And then my slope, which is that m, is three.

00:41 So that's three over one.

00:43 So it's either up three to the right one or down three to the left one.

00:47 So that's what i'm going to do.

00:48 One, two, three, one, two, three one.

00:52 And then this is a dotted line because it's just greater than.

00:57 It's not greater than or equal to.

00:58 So we have boom, boom, boom, there we go.

01:06 Excellent.

01:08 And then my second inequality, y is less than 3x minus four.

01:13 I'm going to do the same thing.

01:14 So my b value here is negative four.

01:18 So one, two, three, four.

01:20 So by the way, i'm using a scale of one here on my coordinate plane.

01:25 And then my m is three.

01:30 So again, three over one, up three.

01:33 One or down three left one and so i'm gonna oh and this is also dotted because it's it's not less than or equal to it's just less than so we're gonna get this nice little dotted line here and something that you may hopefully notice is that these have the same slope for both of these inequalities the slope is three and so that is going to present us with two options.

02:05 We are either going to have points that do satisfy both, or we're going to have no points that satisfy both.

02:13 And that doesn't happen when you have intersecting lines.

02:18 When you have intersecting lines, then you always have one of the quadrants, which will produce solutions for both.

02:27 Because like if you think about it like this, when you have the like let's do i don't know here we go so like if i have a line that does this and then i have a line that does this so let's say like my shading could be here and then it could be here so i've always got this quadrant or even if i put it over here i've got this quadrant if i changed this direction, i put it here.

03:07 I just did that one...

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