In Exercises 1–26, graph each inequality. y≤(1/3)x

Question

Accepted Answer

Hey everyone in this problem, we are asked to graph the following inequality were given, why is less than or equal to 1/6 x. We're gonna go down to the bottom here to give ourselves some more room to work and we're gonna rewrite that inequality, why is less than or equal to 1/6 x. Now when we're giving it inequality like this and we're asked to graph it, the first thing we wanna do is go ahead and graph the equality. Okay, that's gonna give us kind of a boundary, it's gonna separate the plane into two portions. And then we can figure out which of those portions of the plane should be included in our graph for our inequality. So we're going to graph the equality why equals 1/ x. And you'll notice this is an equation of a line, it's written in the form Y equals mx plus B. And the coefficient in front of the X is going to give us our slope. So this is going to be a line with slope 1/6. And we're gonna find our X. And Y intercept. That's just going to help us graph this line. So our X intercept This is gonna be when Y is equal to zero and then y is equal to zero. We get zero is equal to 1/6 times X, which tells us that X is equal to zero. And for our X intercept, this occurs when X is zero. And now we just found that when x is zero, y is zero. Okay, so we expect that to be the same and it is So our x and y intercept occur at the same 00.0. Okay. So we have one point and a slope. So we can go ahead and graph our line. If you want to find another point, you can go ahead and do that as well. So if you plugged in say X equals one, you're gonna get y is equal to 1/6. So you have the 0.1 1/6. Okay. So if you want another point to graph with we have that point. So drawing our graph We have 00, the X&Y intercepts and then we have the .1 and 16. It's going to be like here we can connect those and draw this line with Slope 1 6. Now, what we need to do is we need to figure out if this line should be drawn as a solid line indicating that we're including the boundary or if it should be drawn as a dotted or dashed line indicating that we're not including this boundary. Our inequality is why is less than or equal to 1/ X. So because we have that equals two, we're going to have a solid line because we want to include this boundary in our inequality. So we're going to draw this as a solid line and it's going to look something like this. Alright. And this is our X axis and our Y axis And our line y equals 166. All right now we need to figure out what side of this line. More included. We want y to be less than or equal to one success. So less than or equal to this line. Where do we have Y values that are less than? Okay, well our Y values get smaller or more negative as we go down. So we want why to be below this line. These are the values that are going to satisfy our inequality. Now if you didn't see that or it's confusing to think about it that way. What you can do is you can also choose a point. Okay, so choose a point on the bottom side of this line. So say we choose the point Um 0 -3. If we choose the 0 - we're going to have negative three. Y. The Y value of negative three is less than or equal to zero. That's a true statement. So that satisfies the inequality. So that point is going to be included in our inequality and at that point is below that line. And so we know that that section is included. Okay, you could do the same with a point above plug it in and you would see that it doesn't satisfy the inequality. Alright, so moving back up to our answer choices. Okay, we've drawn our diagram, we just have to find the one that matches. Alright, so we drew our boundary as a line. 1/6 X Y equals 16 X. This is a positive slope. So we're looking at option either C or B. K. The other to have a negative slope. Those don't look like the lines we drew. And then the only difference between CMB is whether we're including above the line or below the line and we found that we needed to include what was below the line and so we have answered. See this matches the graph that we drew below and that's it. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 1–26, graph each inequality. y≤(1/3)x

Key Concepts

Inequalities

Graphing Linear Equations

Shading Regions in Graphs

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