This week in Pre-Calculus 11 we started a new unit which is rational expressions and equations. We first looked into equivalent rational expressions and determining the non-permissible values. Plus, simplifying, dividing/multiplying, and adding/subtracting rational expressions. We also had to review factoring again in order to simplify the rational expressions and find the domain values of the equation.

Since the denominator of the expression cannot be equal to 0, you have to find all of the values of “x” that will result in the denominator to equal to 0. These are called the non-permissible values.

Here is an example of determining non-permissible values of a rational expression:

First you have to factor the bottom, so that you can find the non-permissible values. It’s much easier to find the values if the expression on the bottom is factored.

This is the expression in the denominator:

     x² – x – 6 

To factor you have to use the product and sum law. So, you have to multiply the coefficient in front of the x²  and the -6 to get -6. Then you have to find the 2 numbers that multiply and have the answer of -6, and add to the middle coefficient which is -1. Those two values are 2 and -3.

(x+2)(x-3) = 0

Now you have to use algebra to find the x values:                                               

The non-permissible values for this expression is 3 and -2. So, the x-values cannot be 3 and -2, if they were substituted in for the x then the denominator would be 0, which would result in an undefined answer.

Here is an example of writing a rational expression in simplest form while also finding the non-permissible values:

First you have to factor both the top and the bottom to find the non-permissible values on the denominator plus it will be easier to simplify the expression:

The non-permissible values for this equation is -2 and 2. So, x cannot equal to -2 and 2.

Now, you have to divide the numerator and denominator by their common factors:

This is the equivalent rational expression and the non-permissible values for it.

Here is another example of simplifying a rational expression:

First, you have to factor the denominator and numerator, so that you can simplify the expression easier and find the non-permissible values.

You can also write “(4 – x)” as -1(x-4) because it creates the common factor of (x – 4).

To find the non-permissible values you have to use algebra:

The non-permissible values for this expression is 1.5 and 4.

Now, to simplify the expression you have to divide the numerator and denominator by the common factors:

Here is the simplified version of the original expression. To see if this expression is equivalent to the original, you pick a random number (but not 1.5 and 4) and substitute it for x on both of the expressions. If both of them have the same answer, then you solved it correctly.

Here is an example:

The original expression:

The equivalent expression:

In both expressions, we can substitute “x” for 2:

The original expression:

The equivalent expression:

Both of them are equal to -6, so they’re both equivalent expressions.