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# Two-variable inequalities from their graphs

CCSS.Math:

## 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 is 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 let's think about its slope if we move if we move two in the X direction if we 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 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 and you could have done this anywhere you could have said hey what happens if I go back for in X so if I went back for if Delta X was negative 4 if Delta X is equal to negative 4 then Delta Y is equal to positive 2 Delta Y would be equal to positive 2 and once again Delta Y over Delta X would be positive 2 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 its negative 1/2 so the equation of that line is y is equal to the slope negative 1/2 X plus the y-intercept minus 2 that's the equation of this line right there now this inequality is 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 as you take 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 one let's say X is equal to 2 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 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 and I know that because they shaded in this whole area up here so the equation or I should say the inequality that fits the graph here below is and I'll do it in I'll do it in a in a bold color is y is greater than or equal to negative 1/2 X 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