HELLO!!!
Mr. Tarrou.
We just got done solving or working on a couple of rational... or excuse me polynomial inequalities.
Today, or at least in this video...excuse me...it is the same day, we are working on
solving rational inequalities.
It is almost exactly the same directions except for this part right here.
We are still going to get our rational equation...or inequalities set equal to zero.
We are going to determine the x intercept by setting the numerator equal to zero.
This is the only step that would be required if you were solving for a rational function.
But when it is an equality you have to do one more step and mark off where it is undefined.
So we are going to set the numerator equal to zero and those are going to be the x intercepts.
That is going to be where this rational function graph crosses the x axis.
Then we are also going to determine, by setting the denominator equal to zero, what makes
the rational function undefined.
These two steps are going give us our critical values that we put on the number line.
That will break up the number line into two, or three, or maybe even four...probably not
two intervals... but three or four intervals that we then have to go basically guess and
check to see which of those intervals actually work to solve the inequality.
So let's see what this looks like, shall we:)
So our first question is going to be 2x+4 over x-7 is greater than 0.
Ok, so very basic rational function.
There is just one fraction greater than zero.
We are going to solve this by setting the top equal to zero, then the bottom equal to
zero.
So we have 2x+4=0.
That is 2x=-4.
By dividing both sides by 2 we get that x is equal to negative two.
Now it is not really equal to negative two.
This is a critical value.
This is not really a solution.
This is a critical value you are going to mark on your number line.
Then...
And this graph, if you were to graph this, it would cross the x axis at negative two.
The numerator equals zero, that means that the entire fraction equals, that means y is
equal to zero, and hence you have an x intercept.
Now the denominator also needs to be checked.
We need to find out where this is undefined, so x-7=0.
So x=7.
This is where this graph is undefined.
Great!
Well now we are going to take these two values, place them on a number line, and then those
two values...
Let's go ahead and do that instead of me just talking.
Negative two, positive seven, and now I am going to check a number...because this is
not an equation, it is an inequality...I need this fraction...this rational function to
be greater than zero.
Basically, how do you divide two numbers and get a number bigger than zero or a positive
answer?
They both have to be positive or they both have to be negative.
That is what this number line is going to help us determine.
I am going to do that by checking something in this interval, let's say -3.
We are going to check anything between -2 and 7.
Of course zero is going to be the easiest one.
Then we are going to check anything bigger than 7...8, 10, 12, 100, who cares.
But it has to be bigger than seven so I am going to pick the lowest number I can and
try and do this in my head.
So we are going to check with these values, each of those three intervals, see which ones
work.
The ones that do work are going to be our solutions.
They are going to be our inequalities, or our inequality answers to this inequality.
I hope you got those steps down.
So lets check negative three.
2 times -3 plus 4 all over negative 3 minus 7, is that going to give us an answer greater
than zero?
Well that is -6 plus 4 over -10.
Yes, when you divide by a negative number, this will be -2 over -10, when you divide
two negative numbers you get a positive answer.
So this interval works.
Nice, how did you like that squeak?!
Ok, we might not be able to see the orange very well anyway.
So let's try yellow.
Now we are going to try zero.
We are going to have, again these are the checks trying to find out which of the intervals
on the number line have solutions to this inequality, so 2(0)+4 over 0-7.
Is that greater than zero?
Well, zero, zero, so we are going to have 4 over...Oh my goodness...
negative anything...
nothing negative is bigger than zero.
Negative numbers are to the left of zero on the number, so this section, this interval
did not work.
Then finally the last interval with 8, 2(8)+4 over 8-7.
Is that greater than zero?
Well that is positive and this is positive.
This is going to be 16 and 4 is 20 over 8-7 which is 1, and the last time I checked 20
was bigger than zero.
So the last interval works.
Ok.
So let's write our final answer to this inequality.
Well, my final solution is x has to be less than negative two or greater than seven.
With interval notation that would be negative infinity up to negative two or from seven
to positive infinity.
To the left of negative two going to negative infinity, right from seven going to positive
infinity.
Awesome, that example is done.
I hope that you understand it.
I want to twist it up just a tiny bit.
What if it were greater than or equal to zero?
Well then all the steps are the same.
Our critical values are going to be the same, but I have to change my answers just a tiny
bit.
Because now zero is accepted.
Now not only do I have to check my intervals which of course are still going to be the
same, but I also have to check the x intercepts.
Because with equals, you know, can I use negative two, not just to the left of negative two
but can I use -2?
Well as long as I don't make this original rational function undefined I can use it.
So -2 when plugged into here is not going to cause this to be undefined.
So x can be less than or equal to negative two.
With interval notation that is a square bracket.
But can I use seven?
Again because this whole thing can be zero, so if I use the value of 7 itself will that
make this entire rational function equal to zero?
No.
If I plug in 7, I am going to do seven minus seven which is zero so this is undefined.
So this half of the answer is still going to remain the same.
So there is a little bit a twist when you have that equal sign with the inequality.
You will also have to check the x intercepts.
Basically you will be able to use all of the x intercepts that do not make the denominator
equal to zero.
WHOOO!!!
ALL RIGHT.
Let's go onto to the next one.
I love this stuff.
Ok!
Get this erased on my old style chalk board.
Let's do one more example.
x-1 over x+5 is less than or equal to 5.
Now, remember the steps that I just erased.
We have to set the numerator and the denominator equal to zero.
So if you are tempted to go that is 5/1 let's cross multiply, you are going to get rid of
some of your answers because one of the steps of solving your rational inequalities is what
makes it zero and what makes it undefined.
If you multiply the x+5 over to the other side you lost the value that makes the rational
function undefined.
So do that.
I know you may not like working with fractions but you cannot be tempted by that.
So what we are going to do is subtract both sides by 5.
You can see that I am leaving a lot of space because we are going to have to put that over
one and find a common denominator so that I have one fraction on the left hand side.
So I am going to multiply the top and bottom by x+5.
And we are going to have to make sure that you catch that negative, excuse me, that is
in front of the five and distribute it through the parenthesis.
We are going to get x-1, -5 times x is -5x, -5 times positive 5 is -25, all over our common
denominator now of x+5.
Ok.
That is going to clean up to be -4x from our like terms, -26... all over x+5 is less than
zero.
So now we are kind of looking like that previous example.
We are going to set the numerator and denominator equal to zero, x intercepts and what makes
it undefined.
So we have -4x-26=0.
That is going to be -4x=26.
When I divide by -4 I am going to get x is equal to, let's see, -6.5.
Then the denominator.
x+5 equals 0, so x is another critical value of x=-5.
Actually x cannot be negative five because that makes the original rational function
but it is still a critical value we have to mark on the number line.
So our number line is going to be split up with -5 and -6.5.
What do we have to do?
Ahh yes, check the intervals So check this side, check this, and check this and we are
going to see which ones pass and fail.
So going back to the original inequality we are going to have...
What is to the left of -6.5?
How about negative seven.
So -7-1 over -7+5 is that less than or equal to 5?
Well that is negative eight over negative two (not -3) is less than or equal to five.
The two negatives cancel out.
8/2, 8 divided by 2, that is 4.
We have 4 is less than 5, so this gives a big old check mark.
Let's try another that is between negative six and negative five like negative six.
So -6-1, again back to the original inequality, over -6+5 is less than or equal to 5.
That is -7 over -1.
The last time I checked 7 is bigger than 5 not less than 5.
This second interval, what I just used was the -6, that did not work.
I have a feeling that the last interval is going to work.
So let's try zero.
That is easy.
Zero is bigger than negative five.
So 0-1 over 0+5 is less than or equal to 5.
The last time I looked -1/5 was less than 5.
Great!
So we have the first interval works, the middle interval did not, and the one on the right
works.
Whoo.
We are so close to the answer.
So our final answer is, whoops, our final answer is x has to be less than or equal to
-6.5 or x has to be greater than...hmm...
It is greater than or greater than or equal to?
I cannot plug in negative five so, down here I can't plug in negative five, so that is
going to be greater than negative five.
Did I just say you can't divide by...
I cannot plug in negative five, because negative five plus five is undefined (can't divide
by zero).
So I cannot use a value or say that a value is a solution if it make the inequality undefined.
Ok.
So this is our answer.
Don't forget with that equal sign you have to check the intervals and the x intercepts.
Just don't use the x intercept that makes the inequality undefined.
That in interval notation is negative infinity to -6.5, with the square bracket to take care
of the equal sign, or greater than -5.
So it goes from -5 to infinity...AND BEYOND!
AWESOME.
Alright, Go Do Your Homework:D