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# Graphing two-variable inequalities

Sal graphs the inequality y<3x+5. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to graph the
inequality y is less than 3x plus 5. So if you give us any x-- and
let me label the axes here. So this is the x-axis. This is the y-axis. So this is saying you
give me an x. So let's say we take x is
equal to 1 right there. 3 times 1 plus 5. So 3 times x plus 5. So 3 times 1 is 3 plus 5 is 8. So one, two, three, four,
five, six, seven, eight. This is saying that y
will be less than 8. y will be less than
3 times 1 plus 5. So the y-values that satisfy
this constraint for that x are going to be all of these
values down here. Let me do it in a
lighter color. It'll be all of these values. For x is equal to 1, it'll be
all the values down here, and it would not include
y is equal to 8. Y has to be less than 8. Now, if we kept doing that, we
would essentially just to graph the line of y is equal to
3x plus 5, but we wouldn't include it. We would just include everything
below it, just like we did right here. So we know how to graph just
y is equal to 3x plus 5. Let me write it over here. So if I were to write y is equal
to 3x plus 5, we would say, OK, 3 is the slope. Slope is equal to 3, and then
5 is the y-intercept. Now, I could just graph the
line, but because that won't be included in the y's that
satisfy this constraint, I'm going to graph it as
a dotted line. So we'll start with the
y-intercept of 5. So one, two, three,
four, five. That's the y-intercept. And the slope is 3. So if you go over to
the 1, you go up 3. So let me do that in that
darker purple color. So it'll look like this. It will look like that. That point would be on it. That point would be on it. If you go back, you're going
to go down by 3. So that point will be on it,
that point, and that point, and I'll just connect the
dots with a dotted line. That dotted line is the graph
of y is equal to 3x plus 5, but we're not going
to include it. So that's why I made it a dotted
line because we want all of the y's that are
less than that. So for any x-- so
you pick an x. Let's say x is equal
to negative 1. If you evaluate 3x plus 5 for
that x, you'd get here. But we only care about
the y's that are strictly less than that. So you don't include the line. It's everything below it. So for any x you pick, it's
going to be below that line. You take the x, go up to that
line and everything below it. So for all of the x's, it's
going to be this entire area. Let me draw it a little
bit neater than that. It's going to be this entire
area that's under the line. I'll do it in this orange. It's a little bit
easier to see. So this entire area under the
line is y is less than 3x.