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### Course: Algebra basics > Unit 4

Lesson 8: Graphing two-variable inequalities# Intro to graphing two-variable inequalities

Learn how to graph two-variable linear inequalities like y≤4x+3. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

- How to I solve compound inequalities?(85 votes)
- It sounds like you're asking about systems of inequalities. Solving multiple inequalities together is covered later in this video & exercise series: https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-through-examples(63 votes)

- Is there any way to find the shaded side easier.(7 votes)
- Draw a little man ⛷ on each line as if it were the side of a mountain. ⛰

The side that's*above*ground is the**greater than**side.`≥ or >`

The side that's*below*ground is the**less than**side.`≤ or <`

You can also try ✈️**airplane arms**and align your own arms with each line.

The side*above*your shoulders is the**greater than**side.`≥ or >`

The side*below*your shoulders is the**less than**side.`≤ or <`

(54 votes)

- If anyone doesn't understand here's a summary: You'll be given two (or more) equations. It at the beginning starts as just finding the slope for each equation. Once you graph the different slopes/lines, you'll fill in the lines in a specific direction. < to the right and > would be to the left. However, if you were to have an equal to more/less (basically < or > with a line under it), you'd make it just a regular line. However though if its just <> that'd be a dotted line. (also please upvote)(12 votes)
- Your info is a little off... If you have the shading reversed.

If the inequality is y>mx+b, then you shade above the line. If the line is a vertical line, then you shade to the right.

If the inequality is y<mx+b, then you shade below the line. If the line is a vertical line, then you shade to the left.

Hope this helps.(8 votes)

- 5x-y is greater than or equal to 5 and y<5(5 votes)
- Fblpn,

5x-y >= 5 and y=5

If you change the first equation to slope y-intercept form

5x-y >= 5 add y to both sides

5x-y+y >= 5+y The y-y = 0 and disappears

5x >= 5+y And subtract 5 from both sides

5x-5 >= y Now reverse the sides and reverse the sign

y <= 5x-5 So we now the slope is 5 and y-intercept is (0,-5)

So graph that line (solid because it is also = to

and shade everything below the line since it is also <

The y<5 can be rewritten as

y=0x+5 So the slope is 0 (a horizontal line) that crosses the y axis at 5.

So graph that line (dashed line because it is not = to)

and shade below the line since it is <

Where the shaded areas overlap, that is your solution.

http://www.khanacademy.org/cs/y5x-5-y5/6275881863479296

I hope that helps make it click for you.(11 votes)

- how do I know to shade above or below the line(6 votes)
- If the inequality has a symbol of greater than or equal to or greater than you shade above the line. If the inequality has a less than or equal to or less than symbol you shade below the line.

HOPE THIS HELPS:)(8 votes)

- Why do you have to put an equal sign in place of the greater and less than signs? And are you supposed to divide or multiply when you have an equation like this -3x-y <-1 ( there is suppose to be a line underneath the less than sign) ?(9 votes)
- The line underneath the greater than or less than sign means less than or equal to and greater than or equal to. Sal did this to show you what this means.

For your second question, you need to divide so you get an x on one side of the equation. For example, if y = 3, than the equation would be -3x-3<=-1. You would then subtract 3 from both sides of the equation to get -3x<=-4. Then, divide both sides by 3 to isolate the x on one side. Since you are dividing by a negative number, reverse the less than or equal to to a GREATER THAN or equal to sign.(1 vote)

- How do I write the slope if the line goes straight up (is vertical) and how do I write it's equation if the y-intercept is not given but I have an x-intercept?(5 votes)
- If the line goes straight up, then the line's equation is in the form x = ? because only the y value changes, the x value
**never**changes. The x intercept is all you need to calculate for the equation because that x value is the**same**x value for every point on the line. So if your x intercept is (5,0) then your line's equation would be x = 5.

Hope this helped :)

Happy holidays!!(9 votes)

- if I have something like y>-3 and the question says to 'graph the inequality in the coordinate plane'. Then what does the -3 signify/refer to when I put this inequality into slop intercept form to graph it......?(4 votes)
- -3 is the y-intercept. There is no slope (coefficient of x) so you know this is a straight horizontal line at -3. Since y>-3, any value above y=-3 would be a solution to the problem.(3 votes)

- how can i now if the equation is >= or just > i dont see the deference especially when we have the graph and we wont the equation(3 votes)
- If the line in the graph is solid then the inequality is >=.

If the line is dashed, then the inequality is just >.

If you watch the entire video, you will see toward the end that Sal changes the solid line into a dashed line since the 2nd example is for >.(3 votes)

- At5:48, why is -x/2 the same thing as -1/2x?(3 votes)
- There is an invisible 1 in front of the x, so -x/2=-1x/2, then you can separate into two fractions, -1/2 *x/1, and dividing by 1 does not change anything, so you end up with -1/2 x.(2 votes)

## Video transcript

Let's graph ourselves
some inequalities. So let's say I had the
inequality y is less than or equal to 4x plus 3. On our xy coordinate plane, we
want to show all the x and y points that satisfy this
condition right here. So a good starting point might
be to break up this less than or equal to, because we
know how to graph y is equal to 4x plus 3. So this thing is the same thing
as y could be less than 4x plus 3, or y could be
equal to 4x plus 3. That's what less than
or equal means. It could be less
than or equal. And the reason why I did that on
this first example problem is because we know how
to graph that. So let's graph that. Try to draw a little bit
neater than that. So that is-- no, that's
not good. So that is my vertical
axis, my y-axis. This is my x-axis,
right there. And then we know the
y-intercept, the y-intercept is 3. So the point 0, 3-- 1, 2,
3-- is on the line. And we know we have
a slope of 4. Which means if we go 1 in the
x-direction, we're going to go up 4 in the y. So 1, 2, 3, 4. So it's going to
be right here. And that's enough
to draw a line. We could even go back
in the x-direction. If we go 1 back in the
x-direction, we're going to go down 4. 1, 2, 3, 4. So that's also going to be
a point on the line. So my best attempt at drawing
this line is going to look something like-- this
is the hardest part. It's going to look something
like that. That is a line. It should be straight. I think you get the idea. That right there is the graph
of y is equal to 4x plus 3. So let's think about what it
means to be less than. So all of these points
satisfy this inequality, but we have more. This is just these
points over here. What about all these where
y ix less than 4x plus 3? So let's think about
what this means. Let's pick up some
values for x. When x is equal to 0,
what does this say? When x is equal to 0, then that
means y is going to be less than 0 plus 3.
y is less than 3. When x is equal to negative 1,
what is this telling us? 4 times negative 1 is negative
4, plus 3 is negative 1. y would be less than negative 1. When x is equal to 1, what
is this telling us? 4 times 1 is 4, plus 3 is 7. So y is going to
be less than 7. So let's at least try
to plot these. So when x is equal to-- let's
plot this one first. When x is equal to 0, y is less than 3. So it's all of these points
here-- that I'm shading in in green-- satisfy that
right there. If I were to look at this one
over here, when x is negative 1, y is less than negative 1. So y has to be all of these
points down here. When x is equal to 1,
y is less than 7. So it's all of these
points down here. And in general, you take any
point x-- let's say you take this point x right there. If you evaluate 4x plus 3,
you're going to get the point on the line. That is that x times 4 plus 3. Now the y's that satisfy it,
it could be equal to that point on the line, or it
could be less than. So it's going to go
below the line. So if you were to do this for
all the possible x's, you would not only get all the
points on this line which we've drawn, you would get all
the points below the line. So now we have graphed
this inequality. It's essentially this line, 4x
plus 3, with all of the area below it shaded. Now, if this was just a less
than, not less than or equal sign, we would not include
the actual line. And the convention to do that is
to actually make the line a dashed line. This is the situation if we were
dealing with just less than 4x plus 3. Because in that situation, this
wouldn't apply, and we would just have that. So the line itself wouldn't have
satisfied it, just the area below it. Let's do one like that. So let's say we have y is
greater than negative x over 2 minus 6. So a good way to start-- the
way I like to start these problems-- is to just graph
this equation right here. So let me just graph-- just for
fun-- let me graph y is equal to-- this is the same
thing as negative 1/2 minus 6. So if we were to graph it, that
is my vertical axis, that is my horizontal axis. And our y-intercept
is negative 6. So 1, 2, 3, 4, 5, 6. So that's my y-intercept. And my slope is negative 1/2. Oh, that should be an x there,
negative 1/2 x minus 6. So my slope is negative 1/2,
which means when I go 2 to the right, I go down 1. So if I go 2 to the right,
I'm going to go down 1. If I go 2 to the left, if
I go negative 2, I'm going to go up 1. So negative 2, up 1. So my line is going
to look like this. My line is going to
look like that. That's my best attempt
at drawing the line. So that's the line
of y is equal to negative 1/2 x minus 6. Now, our inequality is not
greater than or equal, it's just greater than negative x
over 2 minus 6, or greater than negative 1/2 x minus 6. So using the same logic as
before, for any x-- so if you take any x, let's say that's
our particular x we want to pick-- if you evaluate negative
x over 2 minus 6, you're going to get that
point right there. You're going to get the
point on the line. But the y's that satisfy this
inequality are the y's greater than that. So it's going to be not that
point-- in fact, you draw an open circle there-- because you
can't include the point of negative 1/2 x minus 6. But it's going to be all the
y's greater than that. That'd be true for any x. You take this x. You evaluate negative 1/2 or
negative x over 2 minus 6, you're going to get this
point over here. The y's that satisfy it are
all the y's above that. So all of the y's that satisfy
this equation, or all of the coordinates that satisfy this
equation, is this entire area above the line. And we're not going to
include the line. So the convention is to make
this line into a dashed line. And let me draw-- I'm trying
my best to turn it into a dashed line. I'll just erase sections of
the line, and hopefully it will look dashed to you. So I'm turning that solid line
into a dashed line to show that it's just a boundary, but
it's not included in the coordinates that satisfy
our inequality. The coordinates that satisfy our
equality are all of this yellow stuff that I'm shading
above the line. Anyway, hopefully you
found that helpful.