3. Final Exam Review:
Eliminate the ‘y’ by adding the two equations.Solve for x, plug back in to find y.
3. Three times the larger of two consecutive odd numbers is
five less that four times the smaller. Find the numbers.
This is from September, you should be able to solve!!
A) 8, 10 B) 15, 17 C) 21, 23 D) 11, 13 E) 8, 9
3(x + 2) = 4x – 5; 3x + 6 = 4x – 5; x = 11
4. Simplifying Rational Expressions
A Rational Expression as a fraction where the
numerator and the denominator are polynomials.
Ex. x²-y²
(x-y)²
To Simplify a rational expression:
1. Factor the numerator & denominator
2. Divide out any common factors
We are working with ratios. Fractions are a type of
ratio, where the part is compared to the whole. All
the rules of fractions still apply, including the
impossibility of zero as a denominator.
5. Simplifying Rational Expressions
Think about it. If the denominator, (the ‘whole’ part
of the fraction) is zero, how can there be a ‘part’ (the
numerator). You can’t have a part of nothing.
If the denominator is zero, there is no problem to
solve, since this is impossible. Therefore, excluded
values mean, “we can solve this problem as long as x
doesn’t make the denominator zero.”
6. Simplifying Rational Expressions
(x - y)(x + y)
(x - y)2
(x-y)² is equal to (x-y)(x-y) so we can
cancel out one of the (x-y)
To simplify we first factor the polynomials, then
cancel any common factors if possible.
x + y
x - y= Simple, yes?
7. Practice 1
3x2
- 4x
2x2
- x
Answer
3x – 4
2x - 1
3x2 - 4x x(3x - 4)
2x2 - x x(2x - 1)
==
Excluded Values: Pay attention and you’ll get it
In the above examples, the excluded values are 0,
𝟏
𝟐
.
Here’s why...
1. When we cancelled the x’s, we divided by x. Since
dividing by zero is undefined, x cannot be zero.
2. Set the denominator equal to zero, and solve for x.
You’ll get 1/2. If x is one-half, the denominator is zero
and you won’t have a problem to solve in the first
place. Thus, x cannot = 1/2
10. Practice 3
What is/are the excluded value(s) in this
expression?
x ≠ -6; division by zero.
x ≠ 6; undefined denominator
11. By now you can see that factoring is an often used
method for simplifying rational expressions.
Sometimes these factors are inverses (their product
= -1) of each other. In this case, you can manipulate
one or the other factor to simplify further.
**Doing this will change the sign of the resulting
fraction.
simplify x2
– 6x + 8
(4 – x)(x + 1)
(x - 4)(x – 2)
(4 – x)(x + 1)
(x - 4)(x – 2)
(4 – x)(x + 1)
(4 - x)(x – 2)
(4 – x)(x + 1)
=
(x – 2)
(x + 1)
12. OK? Good. Complete the class
work & submit tomorrow.
Last Practice; Simplify
2
2
4 4
4
x x
x
2 2
2 2
x x
x x
x 2
2 x
