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## Algebra (all content)

### Course: Algebra (all content)ย >ย Unit 6

Lesson 3: Graphing two-variable inequalities- Intro to graphing two-variable inequalities
- Graphing two-variable inequalities
- Graphs of inequalities
- Two-variable inequalities from their graphs
- Two-variable inequalities from their graphs
- Intro to graphing systems of inequalities
- Graphing systems of inequalities
- Graphing two-variable inequalities (old)
- Systems of inequalities graphs
- Graphing inequalities (x-y plane) review

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# Graphing two-variable inequalities (old)

An old video where Sal graphs the inequality y-4x<-3. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- I dont get the whole concept on inequalities and the filled dot and unfilled dot if anyone could answer my question it would be really helpful(10 votes)
- The closed dot is on an equal to, it is including that point. An unfilled dot is in an less than or equal to, it is not including it.(9 votes)

- At1:38what is a boundry?(7 votes)
- A boundary is the collection of all points of a given set having the property that every neighborhood of each point contains points in the set and in the complement of the set.(5 votes)

- I really don't get the concept of with the < and > and I have HW on this and I am really confused which half should I shade in. Help please(2 votes)
- Basically, if y > 2x+3, you graph:

y = 2x + 3

Then, you have to decide 2 things:`1) dotted line or solid line`

If the sign is < or >, use dotted line

if the sign is <= or >=, use solid line`2) Shade above or below`

if y>something or y >= something, shade above

if y<something or y <= something, shade below(6 votes)

- At4:36do you need a boundry line?(6 votes)
- No because that point was just an example in the shaded region to support the equation.(0 votes)

- how do u graph multiple inequalities(3 votes)
- See the https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-through-examples video & exercise series for dealing with that.(1 vote)

- How do yo know to shade to the left or to the right? If it's less than wouldn't it be to shaded to the left?(1 vote)
- Armin,

If it is less than, shade below. If it is more then shade above.

If you think of it as left and right, then the answer changes when the slope is positive or negative. This makes using left and right much harder to remember.

It is best to think of "less than" as below and "greater than" as above.

The is one exception: If it is a vertical line such as x<1 then there is no above or below. If the line is vertical, you would shade "less than" to the left and "greater than" to the right.

I hope that helps.(5 votes)

- So, do you ever need to solve the inequality before graphing it? Or, do you just graph it anyway like you did in the video?(3 votes)
- Great question. You should always be able to graph it as an inequality. You really can solve it before graphing, but that is up to you most of the time. Make sure that what you do is what the question asks you to do. I hope this helps!(1 vote)

- no the "y" value doesnt change unless you are dividing or multiplying by negatives hope i helped :)(3 votes)
- Ok, I can draw the boundary line and I understand the difference between dotted and solid boundary lines. But how do i know which side to shade? Please be detailed and thanks in advance.(2 votes)
- Where would you use this(2 votes)
- Some jobs like nurses, management people, professors, mostly people who have computers as their jobs, etc have to know what linear inequalities are.(2 votes)

## Video transcript

Graph the inequality y minus
4x is less than negative 3. So the first thing
we could do is we could kind of put
this in mx plus b form, or slope-intercept form,
but as an inequality. So we're starting with y minus
4x is less than negative 3. We can add 4x to both
sides of this inequality. So let's add 4x to both
sides of this inequality, and then we'll just have
a y on the left-hand side. These guys cancel out. So you have y is
less than 4x minus 3. We could have had
negative 3 plus 4x, but we want to write the 4x
first just because that's a form that we're
more familiar with. So it's less than 4x minus 3. And now we can
attempt to graph it. But before I graph it, I want
to be a little bit careful here. So let me draw our axes. So this is the x-axis,
and is that is the y-axis. And we want to be
careful, because this says y is less than 4x
minus 3, not less than or equal to 4x minus 3, or
not y is equal to 4x minus 3. So what we want to do is kind
of create a boundary at y is equal to 4x minus 3, and
the solution to this inequality will be all of the area below
that, all of the y values less than that. So let's try to do it. So the boundary line would look
like-- so let me write it over here-- so we have a boundary
at y is equal to 4x minus 3. Notice this isn't
part of the solution. This isn't less than or equal. It's just less than. But this will at
least help us draw, essentially, the boundary. So we could do it two ways. If you know slope
and y-intercept, you know that 4 is our
slope and that negative 3 is our y-intercept. Or you can literally
just take two points, and that'll help you
define a line here. So you could say, well, when
x is equal to 0, what is y? You get 4 times 0 minus 3, you
get y is equal to negative 3. And we knew that because
it was the y-intercept. So you have 0, and then you
have 1, 2, 0, and negative 3. And then you have
the point, let's say, when x is equal
to-- I don't know-- let's say when x is equal to 2. When x is equal to 2, what is y? We have 4 times
2 is 8 minus 3, y is then going to be equal to 5. So then you go 1, 2, and
you go 1, 2, 3, 4, 5. And so you have that
point there as well. And then we can just
connect the dots. Or you could say, look,
there's a slope of 4. So every time we move
over 1, every time we move 1 in the x direction, we
move up 4 in the y direction. So we could draw it like that. So the line will look
something like this. And I'm just going to
draw it in a dotted line because, remember, this
isn't part of the solution. Actually, let me draw
it a little bit neater because that point should
be right about there, and this point should
be right about there. And then this boundary line I'm
going to draw as a dotted line. So it's going to look
something like that. I draw it a dotted line
to show that it's not part of the solution. Our solution has the
y's less than that. So for any x, so you pick an x
here, if you took 4x minus 3, you're going to
end up on the line. But we don't want the y's
that are equal to that line. We want for that
particular x, the y's that are less than the line. So it's going to be all
of this area over here. We're less than the line, and
we're not including the line, and that's why I put
a dotted line here. You can also try values out. You can say, well, this line
is dividing our coordinate axes into, essentially,
the region above it and the region below it,
and you can test it out. Let's take something
that's above it. Let's take the
point 0, 0 and see if that satisfies
our inequality. If we have y is 0 is
less than 0 minus 3, or we get 0 is less
than negative 3. This is definitely not the case. This is not true. And it makes sense
because that 0, 0 is not part of the solution. Now, we could go on the other
side of our boundary line. And we could take the
point, I don't know, let's take the point 3 comma 0. So let's say that this is the
point-- well, that's right. There's a point 2 comma 0. Let's take the point 3
comma 0 right over here. This should work because
it's in the region less than. But let's verify
it for ourselves. So we have y is 0. 0 is less than 4
times 3 minus 3. 0 is less than 12 minus 3. 0 Is less than 9, which
is definitely true. So that point does
satisfy the inequality. So in general, you
want to kind of look at this as an equal to
draw the boundary line. We did it. But we drew it as a
dotted line because we don't want to include
it because this isn't less than or equal to. It's just less than. And then our solution
to the inequality will be the region
below it, all the y's less than the line
for x minus 3.