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?
And here we got -1/3 is greater than or equal to X
Here we got X is less than or equal to negative 1/3
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.