Pre-Calc 11: Rational Expressions - Chapter 6

This is a big topic that is really important for Pre-Calc 12. Take your time. Ask for help, please!

A review …

Rational expressions is just a fancy name for fractions. In grade 11, fractions take on a bit of a

twist with adding variables in both the numerator and the denominator. It sounds harder than it

is as grade 11 fractions almost always have some sort of cancelling out to happen to make the

fraction simpler.

For success in this chapter, it is important to review two things.

Basic operations with fractions. If you feel like you need a quick review, please see this video.

He does everything that you need for this course, including cross cancelling (which is a big deal

this year and next) as well as not changing to mixed numbers (completely not useful in most

math courses).

Time for some practice: Please go here. You will need to check off the boxes for

adding/subtracting and multiplying/dividing fractions before beginning. You may also wish to

increase the time limit of 20 seconds.

The other topic we have been reviewing the last two chapters – factoring. I haven’t added any

other review here as you have done some over the last few weeks. If you feel like you could use

some review, please refer back to the previous chapters.

6.1 Rational Expressions

Rational expressions is just a fancy phrase for fractions. The twist in grade 11 is that the fractions

will also now have variables in them. The other part to know is that when you see the word

expression, it means to simplify.

Non-permissible values. These refer to numbers that you are not allowed to have. When we are

using fractions, it means that the denominator cannot equal zero because in our math system, we

cannot divide by zero. Non-permissible values are only found in the denominator. The video

for the most basic non-permissible values is here. Please also watch this video as it shows you the

next level for non-permissible values.

Simplifying rational expressions means to look for some possible canceling. Sometimes you will

need to do some factoring first before any cancelling can occur. Remember that it has to look

the exact same before it can be cancelled. As well, you will need to be able to state non-

permissible values any time there is a variable in the denominator. Please view this video. The

only part that he doesn’t do is to state the non-permissible values.

EXTENSION

Sometimes a rational expression will look very close but not identical in the numerator and

denominator. You can try to use the additive inverse property to see if anything in the

numerator and denominator can be cancelled. This video explains both the idea behind the

additive inverse property and gives an example using a variable. If you just want to view the

example using a variable, start the video at 1:18.

Please follow the worked examples below:

3−𝑏

𝑏−3

𝑥2−81

9−𝑥

12−3𝑥

𝑥2+𝑥−20

Time for some practice

Basic: p 317 2, 4, 6 ab, 8 acf

Extension: p 317 2, 4, 6, 8, 9, 13, 22, 25

6.2 Multiplying and Dividing Rational Expressions

Multiplying and dividing rational expressions are a lot like multiplying and dividing fractions.

You do not need a common denominator. When multiplying, cross cancel if you can and then

multiply top by top and bottom by bottom. For dividing, remember to multiply by the

reciprocal of the second term. The only thing that you need to add is to state the non-

permissible values (NPVS).

NPVS have to be stated every step …. 1. Factor and state NPVS

2. Take reciprocal (if dividing) and state NPVS

3. Cross-cancel and check for NPVS

The first video gives an example for multiplying an expression. He does take an extra step and

rewrite the factoring part to line terms out to cancel. He also talks about stating the domain

which is the same as stating restrictions. I don’t normally do that but if it makes it easier for you,

please follow along. The second video gives an example for dividing a rational expression. His

NPVS are correct but more complicated than we need to write. Write the same way as in the

multiplication problem. Be careful if there is a variable in the numerator of the second term. It

will also need a NPVS stated. The last video has a wider variety of examples but does not state

the NPVS. Please remember that these are important in our course.

EXTENSION

This last video gives an example with three terms. Really not much more to think about except

for order of operations. As well, she doesn’t state the NPVS. She makes a mistake part way

through the video. Can you catch it? She catches it before she ends and shows what should have

happened.

Time for some practice

Basic: p 327 2ab, 4, 8ab, 9

Extension: p 327 2, 4, 8, 9, 10, 14

6.3 Adding and Subtracting Rational Expressions

Recall that when adding and subtracting fractions, you must have a common denominator and

then you just keep the denominator and add or subtract the numerators. Keep in mind what

happens when you subtract. You need to make sure that the negative is distributed through the

entire of the second term. He gives some great advice to not miss the negative sign. It is the

same advice I would give you in class. Please note that he does not state NPVS. Please view the

first video here.

The second video covers a slightly more difficult concept. This is finding the common

denominator when the denominators are different. There is a pattern, however …. Please

watch this video. Again, this video does not state the NPVS. Again, he gives some great advice

in finding the LCD and to not FOIL. Remember that if you keep factors, you can do some cross

cancelling.

The last video shows another subtracting example but with more work to do in the denominator

to find common denominators. Again, NPVS aren’t stated in the video.

EXTENSION

The extension for this section is truly more of the work from above with a couple of twists.

Instead of finding a video, I chose to do some examples below.

𝑥−1

𝑥2+5𝑥−6−

𝑥−2

𝑥2+4𝑥+3

3

2𝑥+2

4

3𝑥−1

𝑥2−3𝑥−18

𝑥2+10𝑥÷

𝑥2−13𝑥+42

𝑥2+3𝑥−70−

𝑥+5

2𝑥+3

Time for some practice

Basic: p 336 1 ace, 5 ace, 6 ace

Extension: p 336 all of the above and 7 ac, 8, 10, 15 bd

6.4 Solving Rational Equations

This one is a tough topic. Give yourself some time to work through it. It becomes very

important if you are choosing to go on to Pre-Calc 12.

Solving rational equations is similar to how we solved algebraic equations in grade 9 and 10

when there were rational numbers involved. The trick is to find the lowest common

denominator (LCM). If you multiply every term by the LCM, then you could get rid of the

denominator and just solve for the numerators. The only added part this year is to check for

extraneous roots. Extraneous roots are answers that usually come from solving an equation

algebraically but either don’t make sense in a word problem or don’t work because of NPVS.

To solve rational equations: 1. Factor denominators

2. State NPVS

3. Multiply every term by the LCD

4. Solve

5. Check for extraneous roots

A string of videos to view … It is the same person and she reviews the information each time.

But, she does talk about restrictions!

Video one – this video shows the first type of problem that we do. The second example is

important to see as it shows what happens when your solution is the same as the restriction.

Video two – this video shows what happens when you get an x2 – remember how to factor a

quadratic?

Video three – the first part of this video is a review of the last video. The important part is the

second example as you have to factor to find a common denominator.

Video four – this video goes over some examples. At the end, she talks about and shows an

example of the difference between solving and simplifying. This is important. Please be careful

when you are doing questions from this unit – when you see an equal sign, you need to solve.

When you see an expression, you need to simplify only.

EXTENSION

This extension section is all of the “real world” problems that would be solved using rational

equations. Not sure that you would spend time outside of school to figure out these problems

but they all can be solved with rational equations. There are three types of these problems; rate

of work, distance (think physics, sort of), and mystery number.

Video one – rate of work. They are set up the same way every time. Makes these problems kind

of nice to solve.

Video two – distance. These are the harder of this set of problems. This video is how I set them

up. Here is a variation of these problems – personally I don’t really like the current types of

problems.

The last type of problem is the mystery number problem. I couldn’t find a video showing this

type of problem so I have included an example below.

Two numbers have a difference of 5. The sum of their reciprocals is 1

6. What are the two

numbers?

Time for some practice

Basic: p 348 1-4 (careful of 3d!)

Extension: p 348 all of the above and 9, 11, 12, 14, 15, 17, 18

If at any time you are not sure about what to do, please email for help and we can arrange a

Zoom call or a phone call, whatever will work to help you!

Please email me for access to the quiz once you are done this topic.