If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Two-variable inequalities from their graphs

Sal is given a graph and he analyzes it to find the two-variable inequality it represents. Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

  • male robot hal style avatar for user Andrew
    PLease help me on how to know when the line is shaded and when it is not shaded? Can you help?
    (23 votes)
    Default Khan Academy avatar avatar for user
    • female robot grace style avatar for user ShortCake
      You can test out if the point works by filling in the x and y values of the equation with any point on the graph. I choose 0,0 because it's so easy. If the test with 0,0 is true, then the shaded region includes the point 0,0, and if it's false, shade the opposite region which doesn't include the point 0,0.
      (15 votes)
  • leaf yellow style avatar for user Mo
    Do you flip the inequality sign when multiplying or dividing by a POSITIVE number as well? Or is this rule only applicable to negative multiplication/devision.
    (12 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user littletonamm2122
    what do the dotted and full lines stand for?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      Dotted (or dashed) lines mean that everything on the line does not count as part of the solution (using the < or > inequalities). Solid (or full) line means that the points on the line do count as part of the solution (using the ≥ or ≤ inequalities). These are related to the closed circles (count) and open circles (does not count) used on the number line if you are one variable instead of two.
      (4 votes)
  • duskpin tree style avatar for user Christie
    At , how did Sal know from the question that the inequality would represent the area above the line and not below it?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • starky tree style avatar for user Athens, Adah
    How do you figure out if its greater than or less than
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user FAITH
    I like totally need help with knowing how to write inequalities by looking at intersecting lines?😣🙊
    (3 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user Tete
      It is kinda hard to explain but I will try here:
      when the line is shaded on the top then it will be a ">" if it is "<" and if it is a dotted line then it will be greater than or equal to or less than or equal to. the number on the y axis would be adding or if it is negative then it will be subtracting. Last to the number moving horizontally will be the denominator and the one moving vertically will be the numerator.

      --Hope this helps :)
      (3 votes)
  • aqualine tree style avatar for user Amber
    Why is the inequality about everything above the line and not about everything below the line?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • duskpin sapling style avatar for user Athena
    sorry I'm really confused on how you got the greater than or less then symbols. if there is a big portion of the graph shaded does that mean it's greater than or less than? I already know if the line when the line is greater than or equal to or less than. when the line is solid its greater than or equal to or when the line is dashed its less than or greater than. if you can make a simpler way to help me remember which inequality to put that would be really helpful thank you
    (2 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user batman
      Having a big portion or small portion shaded does not regard whether the symbol has to be greater than or less than.
      Think of it this way: If the shaded part is above the line drawn, the sign is greater than/greater than or equal to. If the shaded part is below, it is less than/ less than or equal to. So phrase it in your mind: "above greater" "below less"
      Hope this helps!
      (3 votes)
  • winston default style avatar for user lev
    What would happen if the shaded area is to the left or right side of the line? Would it be < or >?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      If the shaded area is to the left of the line, the inequality can be written in the form x < something for a dashed line, or the form x <= something for a solid line, where the “something” is a constant or an expression using the variable y only. In the previous sentence, if we replace “left” by “right”, then we replace < by >, and <= by >=.
      (1 vote)
  • female robot ada style avatar for user ananya.jambhale
    How do you get the inequality from y=mx+b?
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

Write an inequality that fits the graph shown below. So here they've graphed a line in red, and the inequality includes this line because it's in bold red. It's not a dashed line. It's going to be all of the area above it. So it's all the area y is going to be greater than or equal to this line. So first we just have to figure out the equation of this line. We can figure out its y-intercept just by looking at it. Its y-intercept is right there. Let me do that in a darker color. Its y-intercept is right there at y is equal to negative 2. That's the point 0, negative 2. So if you think about this line, if you think about its equation as being of the form y is equal to mx plus b in slope-intercept form, we figured out b is equal to negative 2. So that is negative 2 right there. And let's think about its slope. If we move 2 in the x-direction, if delta x is equal to 2, if our change in x is positive 2, what is our change in y? Our change in y is equal to negative 1. Slope, or this m, is equal to change in y over change in x, which is equal to, in this case, negative 1 over 2, or negative 1/2. And just to reinforce, you could have done this anywhere. You could have said, hey, what happens if I go back 4 in x? So if I went back 4, if delta x was negative 4, if delta x is equal to negative 4, then delta y is equal to positive 2. And once again, delta y over delta x would be positive 2 over negative 4, which is also negative 1/2. I just want to reinforce that it's not dependent on how far I move along in x or whether I go forward or backward. You're always going to get or you should always get, the same slope. It's negative 1/2. So the equation of that line is y is equal to the slope, negative 1/2x, plus the y-intercept, minus 2. That's the equation of this line right there. Now, this inequality includes that line and everything above it for any x value. Let's say x is equal to 1. This line will tell us-- well, let's take this point so we get to an integer. Let's say that x is equal to 2. Let me get rid of that 1. When x is equal to 2, this value is going to give us negative 1/2 times 2, which is negative 1, minus 2, is going to give us negative 3. But this inequality isn't just y is equal to negative 3. y would be negative 3 or all of the values greater than negative 3. I know that, because they shaded in this whole area up here. So the equation, or, as I should say, the inequality that fits the graph here below is-- and I'll do it in a bold color-- is y is greater than or equal to negative 1/2x minus 2. That is the inequality that is depicted in this graph, where this is just the line, but we want all of the area above and equal to the line. So that's what we have for the inequality.