Week 13-15

When solving rational equations do I still need to worry about values that cannot be used?

Yes, even if you simplify out certain expressions from the beginning. It is a good idea to figure out what values that are off limits right away, so that you know you can't use them later on.

Is there a way that I can get rid of all the fractions with one step?

Yes, if you multiply the numerator expression and denominator expression by the least common denominator of all the fractions in both the top an bottom you will effectively get rid of all fractions and just have a regular rational expression to compute with.

When simplifying rational expressions do I need to tell people what values can't be used?

Yes, you'll need to explain that the variables in the denominator cannot equal certain values because of the issue of making the denominator zero. You can use the symbol to the left to communicate what values the variable cannot equal.

What happens when you multiply by 1?

1 is a unique number, where anytime you multiply or divide by 1 the original number doesn't change.

What values make a rational expression undefined?

All values that would result in the denominator being zero. This is the job of the person using the expression to define when the expression isn't valid. For example if (x-3) was the expression in the denominator of a fraction, you would have to state that x cannot equal 3 because that would result in the denominator being zero.

Why can't you have a zero in the denominator of a fraction?

If a fraction is considered a quotient, then having a zero as the denominator would be asking the question: Take an amount, and share it equally into no groups. That operation isn't possible, because you cannot have something and then divide it into nothing. Mathematicians have agreed to call this an undefined quantity.

What does all this simplification of rational expressions and complex rational expressions get me?

It will help you solve equations that have rational expressions in them.

How do you divide rational expressions?

Just like the procedure for dividing fractions, you can use the inverse relationship of multiplication. So, flip the second rational expression and multiply to get the answer. If the directions ask for it, make sure to simplify your product to its simplest terms.

How do you add and subtract rational expressions?

Just like you would add or subtract regular fractions. Get denominators to be common, and then combine the numerators. Do you see the multiplication by fancy 1s?

How do you multiply rational expressions?

Just like you would multiply regular fractions. Multiply numerators with numerators, and denominators with denominators. Once finished, if the directions ask for it you should try and reduce your product to its simplest form.

What do the steps of solving a rational equation look like?

Just one example. You can really be imaginative with how to go about solving these equations, sometimes you will not be successful with what you are trying... go back to where you know you were good and try something else.

What are rational numbers?

Rational numbers are fractions or also referred to as the division or quotient of integers. This means that they can be positive or negative.

How do you find what makes the denominator zero and therefore where the expression is undefined?

Set the denominator equal to zero and find all of the answers to that equation that you have created. This could involve factoring, algebra, or many of the other techniques already learned in this course.

What is a rational expression?

This is a fraction or a quotient that involves variables and expressions. Rational or fractional expressions can be thought of as ratios or quotients. You can perform operations (add, subtract, multiply, divide) on them similarly to rational numbers (fractions).

What about fancy 1s, when will that be helpful when dealing with algebra and rational expressions?

This will help you get common denominators so that you can add and subtract rational expressions. Multiplying by something like (x+3)/(x+3) is actually multiplying by a fancy 1. This will allow you to produce equivalent expressions and find common denominators for computation.

Can't I just combine the numerators first, then combine denominators second? Wouldn't that give me an easier complex rational?

Yes! Once you have done that you can follow the procedures for dividing rational expressions to continue the simplifying process.

Can you have fractions that have fractions in the numerator and denominator?

Yes. Think about taking 1/2 and dividing it by 6. In fraction form, that would be a 1/2 in the numerator and a 6 in the denominator.

Applications of Rational Equations: Problems involving motion

You have many examples where using the formula distance = rate x time that will involve have to set up rational equations to find missing values.