Hi, I’m Rob. Welcome to Math Antics!
If you been watching our Algebra Basics series,
you’ve learned a lot about equations so far.
But remember, equations aren’t the only type of comparison in math.
There are also Inequalities like the ones you’ve learned about in our video “Basic Inequalities”.
Those sorts of comparisons use the greater-than and less-than signs,
along with the versions of those that include the equal sign.
And in this video, I’m going to show you
what happens when we use those signs in algebraic equations,
like the equation of a line: y = mx + b.
What would it mean to have something like y > mx + b instead?
To answer that question, let’s first start with the very simple equation: y = x.
If you watched our video called “Basic Linear Functions” (and I highly recommend that you do)
you’ll remember that y = x is just a special case of the Slope-Intercept Form of a line
where m (the slope) equals 1 and b (the y-intercept) equals 0.
This equation tells us that whatever value x is, y must be the same:
if x is 0 then y is 0,
if x is 1 then y is 1,
if x is 2 then y is 2.
Every ordered pair that’s a solutions for this equation
will have the same number for y that it does for x.
And if we graph some of those points on the coordinate plane
(connecting the dots to show that there are infinite possibilities)
we get a diagonal line that goes right through the middle of the quadrants I and III.
Now, what if we change this from the equation y equals x,
to the inequality y is greater-than-or-equal-to x?
How would that effect our graph?
To figure that out, we need to determine all the combinations of x and y
that would make this inequality true.
Well fortunately, all of the solutions we found for y = x will still work for this inequality
because it also contains the equal sign.
That means that the diagonal line can stay on our graph.
But the greater-than sign opens up a lot of new possibilities that would also make this statement true.
For example, if we let x be 1, y could still be 1,
but it could also be any number that’s greater than 1…
…like y could be 2, or 3, or 4 just to name a few possibilities.
Let’s make ordered pairs out of those solutions and set them aside for a second.
Then let’s ask, "what solutions are possible if we let x be 0 instead?"
Again, y could still be the same as x (0) but it could also be any value greater than 0…
y could be 1 or 2 or 3 and so on.
Let’s make order pairs from those also.
And then let's ask, "what are some possibilities if we let x be -1?"
In that case, y could be -1 or anything greater-than -1,
like 0 or 1 or 2 and so on.
Now let’s take these new solutions we’ve found and graph them on the coordinate plane.
Do you notice that all of these new points are above the diagonal line
and none of them are below?
…any guess as to why?
Yep, it’s because on the coordinate plane,
y-values increase as you go up and decrease as you go down.
This diagonal is the y = x line, right?
Every point on the line has a ‘y value’ that’s exactly the same as its ‘x value'.
That means, that every point above the line has a ‘y value’ that’s greater-than its ‘x value’,
while every point below the line has a ‘y value’ that’s less-than its ‘x value’.
In other words, the solutions to this inequality aren’t limited
to just the handful of ordered pairs we identified so far.
ANY point above the y = x line would make the inequality true,
but NONE of the points below it will.
To show that fact on our graph, since we can’t draw an infinite number of points,
the convention is to just shade in that area to cover all of them.
Now, that’s pretty easy to do with computer graphics,
but there are also ways to do it on paper with a pen or pencil.
With a pencil you can shade in large areas lightly by rubbing with the side of the pencil like this.
With a pen, you can use a series of lines or hatch marks
to suggest that the whole area is included in the graph.
Getting back to our digital graph, as you might expect,
if our inequality had used the less-than-or-equal-to sign instead,
then all of the solutions would be on or below the diagonal line like this.
Because both of these inequalities include the equal sign,
any point occurring directly on the y = x line is included in the set of valid answers.
In fact, you can think of these inequalities like combinations
of the equation y = x and the inequalities y > x and y < x.
So what if we got rid of those equal signs
and just used the pure greater-than and less-than signs instead?
In those cases, none of the points exactly on the y=x line
would be a solution to the inequality,
but how can we show that on our graph?
Well, the convention is to draw a solid line when the points exactly on the line are included,
but draw a dashed line when they are not included.
Basically, the line serves as a boundary.
On one side of it, all the points are definitely in the solution set,
and on the other side, all of the points are definitely outside the solutions set.
As for the boundary itself, the question becomes,
“Is the line in or out?”
Yes! Lines are out. My point.
What are you talking about?! Lines are in! That’s my point!
Lines are out! Everyone knows that.
Nuh Uh!
Well if you’re so sure, just look it up online.
Alright, I will!
See… lines are in.
I suppose you believe everything on the internet.
Fortunately, the rules for inequalities are easy to remember:
If the inequality includes the equal sign,
then the boundary line is in the solution set and and you graph it with a solid line.
But if it does not include the equal sign,
then you draw a dashed line to show that the boundary line is not in the solution set.
Okay, so these four graphs should give you a pretty good idea
of how graphing inequalities works in general.
But now we need to go beyond the simple y = x equation
to the general equation of a line.
To do that, I’ll walk you through the process
of graphing the linear inequality: y < 2x - 3
That will involve 3 main steps:
Graphing the boundary line.
Picking a test point.
And shading the proper side of the line.
Let’s start with graphing the boundary line.
To do that, first pretend that the inequality you have is actually an equation
so that you can graph it exactly the same way you’d graph a line.
That means re-writing it as y = 2x - 3.
To graph that line, we just need to plot any two points along it and connect the dots.
It’s kinda like making a function table,
but we only need to find the ‘y output’ values for two different ‘x input’ values.
Any two points will do so I’m going to pick x = 2 and x = -2 to keep things simple.
If x is 2, then y will be 1.
And if x is -2 then y will be -7.
Plotting those two points on the graph and then connecting the dots gives us this line,
which is the boundary line of the inequality.
But, are the points exactly along that line included in the solution set?
To answer that we need to check the inequality sign of the original problem.
In this case, since we have the less-than sign that does NOT include the equal sign,
the line itself won’t be included in the set of valid answers
so we’ll draw it with a dashed line instead of a solid line.
Great! Now that we have our boundary line correct,
we need to figure out which side of the line is in the answer set.
To do that we need to pick a “test point”.
Because we know that all the points on one side of the line are in the answer set,
and all the points on the other side are not,
we only need to know the status of one point to tell which side is which.
In other words, if we can determine whether the test point is in or out,
we’ll know if that entire side is in or out.
It doesn’t matter where the test point is so feel free to pick something really easy to work with.
I mean… why pick something like (-2.17, 3.59)
when you could just pick (0, 0) or (1, 1).
The only real restriction is that the test point can’t be on the boundary line itself.
Once you’ve decided which test point to try,
plug its x and y values into the inequality and then simplify it.
For example, if we pick the point (0, 0) we’d plug those x and y values into the inequality
to get 0 < (2 times 0 minus 3), which simplifies to 0 < -3
But hold on a second… that’s not true! 0 is NOT less than -3.
Ah! …that’s why it’s called a “test point”.
The point that you choose can either pass or fail the test.
In this case, since plugging the point’s x and y values into the inequality gave us a false result,
that means that the point is NOT a valid answer for the inequality.
Therefore, (0, 0) must be on the side of the boundary line
where NONE of the points are included in the solution set,
which means ALL of the points on the other side WILL be included.
So that brings us to the last step.
We just need to shade correct side of the line, like so.
To summarize, if the test point you pick makes the inequality true,
then you’ve picked a point that’s in the answer set
which means that you’ll shade the same side of the graph that the test point is on.
But if the test point gives you a false statement,
then you know that it’s not in the answer set
which means that you’ll leave that side blank and shade the other side.
So, that’s the basics of graphing inequalities,
but there’s a couple more things that you’ll need to know to be successful at it.
After-all, not every inequality that you encounter
will be in a nice, easy to work with form like: y > mx + b.
You may get things that need to be simplified.
Fortunately, you can solve or simplify inequalities
almost the same way that you do with equations.
All of the same principles apply, like combining
like combining 'like' terms'
and the idea that anything you do to one side
needs to be done to the other side too.
However, there are couple situations where you need to flip the inequality sign.
You never had to worry about that with equations
because order doesn’t matter when both sides have the exact same value.
But inequalities tell us that one side has a greater value than the other
and the open end of the inequality sign always faces that greater side.
That means that if you need to switch the left and right sides of an inequality,
you also need to switch the inequality sign.
For example, if you have the inequality 2x < y
and you wanted to rearrange so that y comes first,
you’d need to flip the inequality sign when you do that.
So just remember; If you switch the sides, switch the inequality sign.
Another case where you need to switch the inequality sign is
whenever you multiply or divide both sides by a negative number or term.
You don’t need to do this for any other operations.
You can add or subtract positive or negative terms from both sides,
and you can multiply or divide both sides by a POSITIVE term without needing to switch the inequality sign.
It’s only when you multiply or divide both sides by a NEGATIVE that you need to flip it.
To understand why that’s necessary, have a look at the simple inequality 2 < 4
and imagine what would happen if you multiplied both sides by -1.
In a previous video about integers, you learned that multiplying by -1
switches a number to the opposite side of the number line and vice verse.
Doing that to two different numbers actually changes their order.
Positive 2 is less than 4, but -2 is greater than -4.
That's why every time you multiply or divide both sides of an inequality by a negative,
you need to switch the inequality sign.
For example, suppose you’re given the inequality, -2y + 1 > x + 5,
and you need to solve it for y before you graph it.
How would you do that?
Well, we want to get y all by itself, but two things are currently being done to it.
It’s being multiplied by -2 and 1 is being added to that term.
Let’s undo those operations using order of operations in reverse.
First, we’ll subtract 1 from both sides.
On the left side, the +1 and the -1 cancel out leaving just -2y,
and on the right side the x + 5 - 1 simplifies to x + 4.
Notice we DIDN’T switch the inequality sign when we did that operation.
It’s still the greater than sign.
But now, we need to get rid of the -2 that’s being multiplied by y.
That can be done by dividing both sides by -2.
Because that will change the sign of both sides, we DO need to flip the inequality sign for this operation.
It was the greater-than sign originally, but now it will become the less-than sign.
On the left, the -2 over -2 cancel leaving positive y all by itself.
And on the right, we divide the entire (x + 4) term by -2
which simplifies to -x/2 - 2.
So the final result is y < -x/2 - 2
Alright… so now you know the basics of inequalities in algebra.
You know how to graph an inequality using the three step process we learned in the first part of the video,
and you also know that there are a couple situations where you need to flip the inequality sign
when solving or simplifying inequalities.
You may have noticed that we only worked with linear inequalities in this video,
but the concepts we learned can all be extended to other algebraic functions
that you may learn about in the future.
All you really need to do is change step one of the graphing procedure
from “graph the boundary line” to “graph the boundary function”.
We covered a lot in their video, so it may help to re-watch it if you’re still a little confused.
And the best way to make sure you really understand inequalities in algebra
is to try some practice problems on your own.
As always, thanks for watching Math Antics, and I’ll see ya next time.
Learn more at www.mathantics.com