Today

Make-Up Tests? After Today's Lesson Final Exam Review

Begin Rational Expressions UnitGentle Reminder: Khan Academy due

SaturdayClass Work

Number Sense: People

1. The planet recently passed what important population number?

7,000,000,000

2. Most experts generally agree on this estimate for the total number of people who have ever lived on Earth.

118,000,000,000

3. What percent of people who have ever lived are alive today?

< 6%

Number Sense: Age

4. How many of the 7+ billion people alive today are > 110 years old 67

5. How many of those are men? 1

6. The oldest person alive today is.......? 116 years

6. The life expectancy for men born in the 1990’s is... 87.5

6. The life expectancy for women born in the 1990’s is... 93 years

Final Exam Review:

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.

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

Class Notes Section of Notebook, pls.

Additional Resources available@ V6math

Today's Goals:1. Recognize Rational Expressions2. Determine the excluded value(s) of each expression3. Utilize your understanding of factoring to simplify

Introduction to Rational Expressions & Equations

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 & denominator2. Divide out any common factors

Rational Expression Examples:

For students, one of the most confusing aspects of this process is knowing what can and cannot be cancelled.

Remember, terms cannot be cancelled, only factors can be cancelled.

2a + 64a - 2

Simplify: NO, CANNOT CANCEL TERMS

2a + 64a - 2

2(a + 3)2(a – 1)

YES!!!

Simplify &

3x2 - 4x2x2 - x

Answer

x(3x - 4)x(2x - 1)

3x2 - 4x2x2 - x = 3x – 4

2x - 1=

Excluded Values must be noted when:1. Dividing (cancelling terms)

2. The final denominator is determined

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.

Excluded Values

Once more, simplify & state the excluded values

Practice 3

What is/are the excluded value(s) in this expression?

x ≠ -6; division by zero.

x ≠ 6; undefined denominator

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)

OK? Good. You should be able to complete Parts I & II of the class work

Last Practice; Simplify

2

2

4 4

4

x x

x

2 2

2 2

x x

x x

x 2

2 x

Your Penultimate Class Work 4.11 & 4.12

~As usual, show all your work for credit

Polynomial?

Simplify

(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 + yx - y= Simple, yes?

Simplifying Rational Expressions

Simplifying Rational Expressions

If the numerator is zero, you have zero parts of the whole. You have nothing. It's ok to have nothing.

If the denominator is zero, there is no problem to solve, since this is impossible. Therefore, excluded valuesmean, “we can solve this problem as long as x doesn’t make the denominator zero.”

Why can the numerator of a fraction be zero but not the denominator?

However, 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.

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.

Practice 2

(By Grouping)