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Graphing systems of inequalities

Sal graphs the solution set of the system "y≥2x+1 and y<2x-5 and x>1.". Created by Sal Khan and Monterey Institute for Technology and Education.

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  • marcimus pink style avatar for user Ackeebasimms
    Do you have an easier way to know which side to shade ?
    (25 votes)
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    • mr pants teal style avatar for user Tim
      My method is to pick a point which will definitely lie on one side or the other (not on the line) and determine if it fits the equation. You can pick a point which is really easy; usually the origin is a good one.

      For example, if we start with:
      7y < (3/2)x + 5
      It seems annoying. Sub in the origin (0,0) and we get:
      0 < 0 + 5, or 0 < 5.
      That is, the xs and ys just disappear! Easy sauce! This is true, (0 is less than 5), so the side with the origin should be shaded.

      Just remember to be careful with sign. For example:
      7y < (3/2)x - 5
      becomes:
      0 < -5
      Obviously false - don't shade this side. But it is easy on a quick glance to forget that 0 is actually more than -5. Sounds silly, but it's one of those silly mistakes I make - a LOT.

      Hope this helps!
      (41 votes)
  • leaf orange style avatar for user Marquez
    Why is my graphing calculator making X>1 different than the way your doing? It's making a line on Y 1. Please help if this makes any sense to anyone who reads this.
    (10 votes)
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  • piceratops tree style avatar for user Megan Strickfaden
    im confused on how you new which way the coordinate of x>1, at about 3:2
    (12 votes)
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    • hopper cool style avatar for user Chuck Towle
      x=1 would be graphed as a vertical line that is on crosses the x axis at 1. Skip the rest of this paragraph if that already clicks for you. If not, you could also think of it as taking any y, the x coordinate =1, so pick any two y such as 2 and 3. Since you know x always equal 1, then you get the two points (1,2) and (1,3). If you graph the line through these two points, You will see that you get the vertical line going through the point (1,0).

      So now since the inequality is > and not greater than or equal to, you use a dashed vertical line.

      And not for what you asked. To figure out which side to shade, when x > 1, you can choose any point where x is greater than 1 such as (3,3) or (2,-1) and graph that point. Since that is a point you want to include, and you see that point is on the right, you would shade the area on the right. After a couple times it will just click that x > any number is a dashed vertical line at that the point (0,that number) shaded on the right.

      I hope that helps.
      (22 votes)
  • aqualine ultimate style avatar for user L.H.Marten
    How do you tell which side of the line that you shade?
    (10 votes)
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    • mr pants teal style avatar for user ishav.shukla37
      For Example:
      y is equal to or GREATER than 2x+1
      since y is greater than the line itself or the points on the line, you would shade up.

      x is equal to or LESS than 1
      since we are talking about s values, we should shade right or left not up or down. Also since x is LESS than one we should shade everything to the left of one because everything to the left of one is less than 1.

      Hope that helps :)
      (13 votes)
  • aqualine ultimate style avatar for user katherine
    Is there a way to solve a system of inequalities without graphing?
    (6 votes)
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  • duskpin sapling style avatar for user RUBY :)
    What if y has a number next to it like for example 3y, but has the other variable without a number...like 3y < -x-1 ....what you do then
    (6 votes)
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  • blobby green style avatar for user Jojo
    So we just memorize what goes on top and bottom? Any tips/ tricks?
    (4 votes)
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    • aqualine ultimate style avatar for user AD Baker
      1015809,

      Yes. Memorize these facts:

      If the inequality is < or > (with no equal to), the line is dashed.

      If the inequality is <= or >= (contains equal to), the line is solid.

      If the inequality is < or <=, shade below the line.

      If the inequality is > or >=, shade above the line.
      (5 votes)
  • male robot donald style avatar for user Bleeblah101
    how do you know if you shade above or below?
    (4 votes)
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    • male robot hal style avatar for user Andrew M
      Try one "test" point and see if it works. If it does, you shade the side that point is on. If it doesn't, you shade the other side. For example, if you have y>5, then if your test point is y =6, you find 6>5, which is true, so you shade that side. If you chose y = 4 for your test point, then you have 4 >5, which is not true, so you shade the other side.
      (6 votes)
  • blobby green style avatar for user Shubhi  Mukesh
    I still don't understand which part of the graph to shade..heellpp!
    (3 votes)
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  • blobby green style avatar for user e2egarita
    How would u graph a problem with the equation of 3x<y
    (2 votes)
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Video transcript

We're asked to determine the solution set of this system, and we actually have three inequalities right here. A good place to start is just to graph the solution sets for each of these inequalities and then see where they overlap. And that's the region of the x, y coordinate plane that will satisfy all of them. So let's first graph y is equal to 2x plus 1, and that includes this line, and then it's all the points greater than that as well. So the y-intercept right here is 1. If x is 0, y is 1, and the slope is 2. If we move forward in the x-direction 1, we move up 2. If we move forward 2, we'll move up 4, just like that. So this graph is going to look something like this. Let me graph a couple more points here just so that I make sure that I'm drawing it reasonably accurately. So it would look something like this. That's the graph of y is equal to 2x plus 1. Now, for y is greater than or equal, or if it's equal or greater than, so we have to put all the region above this. For any x, 2x plus 1 will be right on the line, but all the y's greater than that are also valid. So the solution set of that first equation is all of this area up here, all of the area above the line, including the line, because it's greater than or equal to. So that's the first inequality right there. Now let's do the second inequality. The second inequality is y is less than 2x minus 5. So if we were to graph 2x minus 5, and something already might jump out at you that these two are parallel to each other. They have the same slope. So 2x minus 5, the y-intercept is negative 5. x is 0, y is negative 1, negative 2, negative 3, negative 4, negative 5. Slope is 2 again. And this is only less than, strictly less than, so we're not going to actually include the line. The slope is 2, so it will look something like that. It has the exact same slope as this other line. So I could draw a bit of a dotted line here if you like, and we're not going to include the dotted line because we're strictly less than. So the solution set for this second inequality is going to be all of the area below the line. For any x, this is 2x minus 5, and we care about the y's that are less than that. So let me shade that in. So before we even get to this last inequality, in order for there to be something that satisfies both of these inequalities, it has to be in both of their solution sets. But as you can see, their solutions sets are completely non-overlapping. There's no point on the x, y plane that is in both of these solution sets. They're separated by this kind of no-man's land between these two parallel lines. So there is actually no solution set. It's actually the null set. There's the empty set. Maybe we could put an empty set like that, two brackets with nothing in it. There's no solution set or the solution set of the system is empty. We could do the x is greater than 1. This is x is equal to 1, so we put a dotted line there because we don't want include that. So it would be all of this stuff. But once again, there's nothing that satisfies all three of these. This area right here satisfies the bottom two. This area up here satisfies the last one and the first one. But there's nothing that satisfies both these top two. Empty set.