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# Systems of Linear Inequalities

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- Everything we've been doing so far has
- involved systems of equations.
- In this video, we're going to tackle the problem of systems
- of linear inequalities.
- What we're going to do is, essentially, figure out all of
- the points on the xy plane that satisfy a system of
- linear inequalities.
- So let's start with our first linear inequality.
- Let's say we have 2x minus 3y is less than or equal to 21.
- And we've seen how to graph this before.
- You can just graph this line, and then figure out what the
- less than part means.
- So let's just rearrange this into slope-intercept form.
- So if you subtract 2x from both sides, we get negative
- 3y-- these cancel out --is less than or
- equal to 21 minus 2x.
- Or we could say negative 2x plus 21.
- And then we can divide both sides by negative 3.
- And remember, when you divide by a negative number, or you
- multiply by a negative number, you swap the inequality.
- So you get negative 3y over negative 3 is greater than or
- equal to negative 2 over negative 3x plus 21 over
- negative 3.
- And so this equation becomes y is greater than or equal to
- 2/3 x minus 7.
- So that's the first inequality right there.
- And let's graph it.
- I'll graph it in orange.
- y-intercept is negative 7, so 1, 2, 3, 4, 5, 6, 7.
- And it's slope is positive 2/3.
- So when you run 3, you go up 2.
- When you run 3, you go up 2.
- And in this direction, if you go back 3, you go down 2, so
- the line will look something like this.
- The line will look like that.
- Keep going.
- And since we have y is greater than, or equal to,
- essentially, this line, the area that satisfies this
- inequality is going to be above the line.
- At any point, for any x you pick, this is the
- value 2/3 x minus 7.
- So if y is going to be greater than that, it'll be all the
- points above that.
- So just this constraint right here, y is greater than 2/3 x
- minus 7, or this constraint, which is the same thing, 2x
- minus 3y is less than or equal to 21, will give you all of
- this area up here.
- I don't want to draw it too dark just yet.
- So it'll give you all of this area all the
- way above this line.
- Now let's add another inequality to the mix.
- Let me do that other inequality in blue.
- Let's say I have x plus 4y.
- Let's make it less than 6.
- The problem I have written here, has less than or equal
- to, but I want to mix it up with an equal sign and a
- couple of them without an equal sign.
- Just so we remember how to graph these things.
- And so let's put this again into slope-intercept form, so
- we can subtract x from both sides.
- Then the left-hand side just becomes 4y.
- The right-hand side becomes a negative x plus 6.
- And then we can divide both sides by 4.
- And we get y is less than negative 1/4 x plus 6.
- So just like the first equation,
- let's graph the line.
- The y-intercept is positive 6.
- So 1, 2, 3, 4, 5, 6.
- And then the slope is a negative 1/4.
- So every time we run by 4, 1, 2, 3, 4, we're
- going to go down 1.
- So you run by 4, 1, 2, 3, 4, you go down 1.
- You run by 4, 1, 2, 3, 4, you're going to go down 1.
- So this line is actually going off the table.
- To go back by 4, 1, 2, 3, 4, you're going to go up by 1.
- And since we're not actually including the line, we're
- going to have to draw a dotted line here.
- This is going to be a dotted line.
- Try my best to draw a dotted line.
- And actually, let me continue that previous line there
- because I actually want show where they intersect.
- Goes a little bit off of our graph paper.
- But the graph of this equation is that right.
- I did a dotted line because it's not less than or equal
- to, it's just less than.
- So this inequality, if I were to graph it, would be the area
- below the line right there.
- That's what satisfies this inequality.
- For any x, this is the value of that line.
- But the y's that satisfy it are the y's that
- are below the line.
- Now if I were to ask you, if I were to have a system, where I
- have 2x minus 3y is less than, or equal to 21.
- These aren't equations.
- These are inequalities, they're not
- equal to each other.
- The other one is x plus 4y is less than 6.
- And if I were to ask you what x's and y's satisfy both of
- these any inequalities, well you say, it's just going to be
- the overlap of their solution sets, right?
- It's going to be the area in the xy plane that satisfies
- both of them.
- So in this case, it'll be the part that's
- above the orange line.
- Remember I shaded that in orange right there.
- So all of this above the orange line, and the part
- that's below the blue line, and the part that overlaps
- that satisfies them both of these-- I'll do it in yellow
- --the part that satisfies both of these is right in between
- the lines there.
- Now what if, just for fun, we were to add a third equation.
- Let's add a third equation in green.
- 3x plus y is greater than or equal to negative 4.
- Well, it's the same drill.
- We put this in the slope-intercept form.
- So you get y is greater than or equal to
- negative 3x minus 4.
- I just subtracted 3x from both sides.
- We can graph this line.
- It's y-intercept is 1, 2, 3, negative 4.
- And it has a slope of negative 3.
- So every time you go one to the right, you go down 3.
- 1, 2, 3.
- If you go one to the left, if you go negative 1 in the
- x-direction, you go up 3.
- So you go negative 1, 1, 2, 3.
- Negative 1, 1, 2, 3.
- Negative 1, 1, 2, 3.
- So this line right there's going to look like this.
- It's going to look like that.
- Trying my best to draw it.
- That's that right there.
- And I can have it in bold.
- I don't have to do a dotted line because I have a greater
- than or equal to and so what are all the y's that are
- greater than this line?
- Well, for any x value you pick, the value of negative 3x
- minus 4 is going to be this.
- And we care about the y values that are greater than that, so
- it'll be above the line.
- So for just this inequality it'll be
- everything above the line.
- And then if someone were to come along and say, all right
- , tell me, show me, graph me, all of the points on the xy
- plane that satisfy this system of inequalities.
- What would you tell them?
- Well, it would be the overlap of their solution sets?
- So the part of the xy plane that exists in all of their
- solutions, so what's above the orange line, right?
- We have to be above the orange line, right?
- And we graphed that before.
- It was all in orange before, because we're above the line
- to 2/3x minus 7.
- It might trick you a little bit because
- there's a less than here.
- But when you put it in the slope-intercept form, you have
- a greater than.
- So it's everything above this line, which is all of that
- over there.
- Everything below this line-- below the blue line and not
- including the blue line --and everything
- above the green line.
- So it's going to be this area right over here, is the
- solution set.
- And we're going to include this boundary right here, and
- we're going to include that boundary right there.
- And since I engineered the problem to have a less than
- there, not a less than or equal to, up here is going to
- be a dotted line.
- Because you're not actually going to include that line.
- Anyway, hopefully you found that useful.

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