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# 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.

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.