Welcome to the presentation on solving inequalities,
or I guess you call them algebra inequalities.
So let's get started.
If I were to tell you that, well, let's just say x is
greater than 5, right?
So x could be 5.01, it could be 5.5, it could be a million.
It just can't be 4, or 3, or 0, or negative 8, and actually,
just for convenience, let's actually draw that
on the number line.
That's the number line.
And if this is 5, x can't be equal to 5, so we draw a big
circle here, and then we would color in all the values
that x could be.
So x could be just the small-- it could be 5.0000001.
It just has to be a little bit bigger than 5, and any of those
would satisfy it, right?
So let's just write some numbers that satisfy.
6 would satisfy it, 10 would satisfy it, 100
would satisfy it.
Now, if I were to multiply, or I guess divide, both sides of
this, I guess we could say, equation or this inequality
by negative 1, I want to understand what happens.
So what's the relation between negative x and negative 5?
When I say what's the relation, is it greater than or is
it less than negative 5?
Well, 6 is a value that works for x.
So negative 6, is that greater than or less than negative 5?
Well, negative 6 is less than negative 5, right?
So let me draw the number line here.
If we have negative 5 here, and let's just draw a circle around
it because we know it's not going to be equal to negative 5
because we're deciding between greater than or less than.
So we're saying 6 works for x.
So negative 6 is here, right?
So negative 6 is less than negative 5, so is negative 10,
so is negative 100, so is negative a million, right?
So it turns out that negative x is less than negative 5.
And this is really all you have to remember when you are
working with inequalities in algebra.
Inequalities, you can treat them just the way-- a greater
than or less than sign, you could treat them exactly the
way you would treat an equal sign.
The only difference is, if you multiply or divide both sides
of the equation by a negative number, you swap it.
That's all you have to remember.
Let's do some problems, and hopefully, that'll
hit the point home.
And if you ever forget, you just have to try-- you just
remember this: if x is greater than 5, well, then negative
x is less than negative 5.
And keep trying out numbers.
That's what's going to give you the best intuition.
Let's do some problems.
So if I said that 3x plus 2 is, let's say, less than or equal
to 1-- well, this is a pretty easy equation to solve.
We just say 3x-- let's subtract 2 from both sides, and when you
add or subtract, you don't do anything to the inequality.
So if you subtract 2 from both sides, you get 3x is less than
or equal to negative 1, right?
And then, now we're going to divide both sides by 3.
We get x is less than or equal to negative 1/3, right?
And notice, we didn't change anything because
we divided both sides by a positive 3, right?
We could have actually done this equation in a
slightly different way.
What if we subtracted 1 from both sides?
So this is another way of solving it.
What if we said 3x plus 1 is less than or equal to 0, right?
I just subtracted 1 from both sides, and now let's subtract
3x from both sides.
And we get 1 is less than or equal to minus 3x, right?
I subtracted 3x from here, so I have to subtract 3x from here.
Now, I would have to divide both sides by a
negative number, right?
Because I'm going to divide both sides by negative 3.
So I get negative 1/3 on this side, and based on what we had
just learned, since we're dividing by a negative number,
we want to swap the inequality, right?
It was less than or equal, now it's going to be
greater than or equal to x.
Now, did we get the same answer when we did it
both-- two different ways?
Here, we got x is less than or equal to negative 1/3, and
here we got negative 1/3 is greater than or equal to x.
Well, that's the same answer, right? x is less than or
equal to negative 1/3.
And that's-- I always find that to be the cool
thing about algebra.
You can tackle the problem in a bunch of different ways, and
you should always get to the right answer as long as
you, I guess, do it right.
Let's do a couple more problems.
Oh, let's erase this thing.
There you go.
I'll do a slightly harder one.
Let's say negative 8x plus 7 is greater than 5x plus 2.
Let's subtract 5x from both sides.
Negative 13x plus 7 is greater than 2.
Now, we could subtract 7 from both sides.
Negative 13x is greater than minus 5.
Now, we're going to divide both sides of this
equation by negative 13.
Well, very easy.
It's just x, and on this side, negative 5 divided by
negative 13 is 5/13, right?
The negatives cancel out.
And since we divided by a negative, we switch the sign.
x is less than 5/13.
And once again, just like the beginning, if you don't believe
me, try out some numbers.
And I remember when I first learned this, I didn't believe
the teacher, so I did try out numbers, and that's how I got
convinced that it actually works.
When you multiply or divide both sides of this equation
by a negative sign, you swap the inequality.
And remember, that's only when you multiply or divide, not
when you add or subtract.
I think that should give you a good idea of how
to do these problems.
There's really not much new here.
You do an inequality-- or I guess you could call this an
inequality equation-- you do it exactly the same way you do a
normal linear equation, the only difference being is if you
multiply or you divide both sides of the equation by a
negative number, then you swap the inequality.
I think you're ready now to try some practice problems.
Have fun.