Algebra: Solving Inequalities

Welcome to the presentation on solving inequalities

or I guess you could call them
algebraic inequalities.

So let's get started.

If I were to tell you
that, well, let's just say

x > 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 -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 5.000001,

it just has to be
a little bit bigger than 5

and any of those
would satisfy, 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 -1,
I wanna understand what happens.

So what's the relation
between -x and -5?

And when I say,
what's the relation,

is it greater than
or is it less than -5?

Well, 6 is a value
that works for x,

so -6, is that greater than
or less than -5?

-6 is less than -5, right?

So let me draw
the number line here.

If we have -5 here--
let's just draw a circle around it

because we know
it's not gonna equal to -5

because we're just deciding

between greater than
or less than.

So we're saying 6 works
for x, so -6 is here, right?

-6.

So -6 is less than -5

So is -10, so is -100, so is -1,000,000, right?

So, turns out -X is less than -5

So 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 > or a < sign you can treat them just the way you would treat an = 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 will bring the point home

If you ever forget, you just have to try, you just have to remember this

X is > 5, well then -X < -5

And keep trying out numbers

That's what is going to give you the best intuition

Let's do some problems

So if I say that 3X + 2 is less than or equal to 1

Well, this is a pretty easy equation to solve

We say 3X, let's subtract 2 from both sides

When you do 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 -1, right?

And then now we are going to divide both sides by 3

We get X is less than or equal to -1/3

Look we didn't change anything

Because we divided both sides by a positive 3

Alright? We could have 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 + 1 is equal to or less than 0

I just subtracted 1 from both sides

And now I'll subtract 3X from both sides

I'll get 1 is less than or equal to -3X

I subtracted 3X from here

So I'll subtract 3X from here

Now I'll have to divide both sides by a negative number

Right? Because I'm going to divide both sides by -3

So I get -1/3 on this side

And based on what we had just learned

Since we are dividing by a negative number

We want to swap the inequality right?

It was less than or equal to

And it's going to be greater than or equal to X

Now did we get the same answer when did both in two different ways?

Here we got X is less than or equal to negative 1/3

And here we got -1/3 is greater than or equal to X

That's the same answer right?

X is less than or equal to negative 1/3

That's what I find to be the cool thing about algebra.

You can tackle a problem in a bunch of two different ways.

You should get the right answer as long as I guess you do it right.

Let's do a couple more problems.

Erase the thing. Here you go. Let's do a slightly harder one.

Let's say - 8x + 7 > 5x +2

Let's subtract 5x from both sides.

- 13x + 7 > 2

Now we can subtract 7
from both sides,

-13x > -5.

Now we're gonna divide both
sides of this equation by -13.

Well, very easy.

It's just x, and on this side
-5/-13 = 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.

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 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 would 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.