Download Ch 11 Alg 1 07

Math is cool!
Especially Chapter
11 in the Algebra 1
book!
Slide Show by Andrew Sublett
and Jeehee Cho
Proportion – an equation that states that two ratios are equal
Reciprocal Property of Proportions
If two ratios are equal, then their reciprocals are also equal.
Ex: If a/b = c/d, then b/a = d/c
EXAMPLE
If you have a proportional equation with a variable such as x, flip both fractions so that
the x is on top.
2/3 = 4/x
First, write down the given equation
3/2 = x/4
Next, flip both sides so that the x is on top
(4)(3/2) = (x/4)(4) Multiply both sides by four to get rid of the
fraction on the x side
(4)(3/2) = x
x=?
x = (12/2)
Multiply
x=6
Simplify
Cross Product Property
The product of two diagonals in a proportion equals the product of the
other two diagonals.
Ex: a/b = c/d ------------- ad = bc
EXAMPLE
5/x = 3/7
First, write the proportion
(5)(7) = (3)(x)
Next, multiply the diagonals
35 = 3x
Multiply out
x = 35/3
Divide both sides by 3
The Cross Product Property can also be used if you have a variable in both fractions
EXAMPLE
2/x = (x+1)/3
First, write the proportion
(2)(3) = (x)(x+1)
Multiply diagonals with cross product property
6 = x^2 + x
Multiply left and right sides (Hint: distribute on right side)
0 = x^2 + x – 6
Subtract 6 from both sides
0 = (x+3)(x-2)
Factor the equation
x = -3, x = 2
Set both (x+3) and (x-2) to zero and solve
Word Problem Example:
Every 6 people use up a total of 12 pencils on a test. If one person leaves the
room, how many pencils will be used up on the test?
Number of people in the room originally = Number of people after one leaves
Number of pencils used initially
Number of pencils used (x)
6/12 = 5/x
Set up the proportions with the given information
6(x) = 12(5)
Multiply diagonals using the cross product property
6x = 60
Multiply out
x = 10 pencils
Divide each side by 6 *hint: don’t forget your units!
A rational expression is a fraction where neither the
numerator nor the denominator are zero.
Example: 2x/34 and ½
When simplifying:
1. Break the expression into simplest terms. Try
factoring out common numbers.
Ex: x^2/ x^3 = (x)(x)/ (x)(x)(x)
2. Cancel out the common terms in the numerator and
denominator
Ex: (x)(x)/(x)(x)(x)
3. Simplify the fraction
Ex: 1/x
*HINTS*
You can only cancel the same amount of variables in
the numerator as in the denominator
 Ex: if you have (x)(x)(x)/ (x)(x)(x)(x) then you
can only cancel out three x’s in the numerator
and denominator to equal 1/x
 You can only cancel out numbers/ variables that are
being multiplied
 Ex: (x^2 + 2x +1)/ (x+1) can be factored into
(x+1)(x+1)/(x+1) and simplify into (x+1)/1 because
the top two are being multiplied
 Ex: (x+2)/x cannot be simplified because the
numerator is simply and addition expression and
cannot be factored

Examples
1.
x^3/ (x^4 + x^3)
x^3/ x^3 (x + 1)
Factor out x^3 in the denominator
Cancel out common x^3 in the numerator
and denominator
1/ (x+1)
2.
Simplify
(x^2 + 6x + 9)/(x^2 + 5x + 6) Factor out both numerator and
denominator
(x + 3)(x +3)/ (x+3)(x +2)
Cancel out the common factor, (x+3)
(x +3)/(x+2)
Simplify
Multiply and divide rational expressions with the same
properties as you would multiply fractions.
Ex: (4x^2/ 3x ) • (5x/ 2x^3)
= (4x^2)(5x)/(3x)(2x^3)
Ex: (3x)/(2x^2) ÷ (3)/(4x)
= (3x)/(2x^2) • (4x)/ (3)
Multiply numerators with
numerators and denominators with
denominators
When dividing fractions, flip the
fraction you are diving by and
multiply out
Multiplying Rational Expressions
(4x^2/ 3x ) • (5x/ 2x^3)
= (4x^2)(5x)/(3x)(2x^3)
Multiply numerators and denominators
*hint* When you multiply exponents, multiply the numbers together first, like 4 and
5, in the numerator, and then the variables such as x^2 and x. For the
denominator, you would multiply the 3 and 2 together and the x and x^3
together.
= (4x^2)(5x)/(3x)(2x^3)
= [(4 • 5)(x^2 • x)]/[(3 • 2)(x • x^3)]
= (20)(x^3)/(6)(x^4)
= 20x^3/ 6x^4
Using rules from the last section about simplifying rational
expressions:
= 10x^3/ 3x^4
Divide numerator and denominator by 2
= 10(x)(x)(x)/3(x)(x)(x)(x) Cancel out common x’s
= 10/3x
Simplify
Dividing Rational Expressions
6x/(4x+1) ÷ 12/(4x+1)
Flip the fraction that you are
dividing by and multiply
6x/(4x+1) • (4x+1)/12
Multiply
6x(4x+1)/12(4x+1)
Cancel out common factors
6x/12
Simplify fraction
x/2
When adding or subtracting rational expressions with the same
denominator, just add the numerators together and your answer will
be one fraction with the common denominator on the bottom and sum
of the numerators on top.
Ex: 3x/(5x+2) + 2x/(5x+2) =
= (3x+2x)/(5x+2)
Add the numerators over the
common denominator
= 5x/(5x+2)
Simplify the numerator
Ex: 2/(7x+1) – (2x+3)/(7x+1)
= (2-(2x+3))/(7x+1)
= (-2x-1)/(7x+1)
Subtract the numerator over the
common denominator (7x+1). *hint*
remember to distribute the
negative sign when subtracting i.e.
(2-(2x+3)) is really (2-2x-3).
Simplify
Least Common Denominator (LCD) – The lowest number that the
denominators in two fractions can each equal by multiplying by any other whole
number.
Ex: 5/6x and 4/36x^2
6x • 6x = 36x^2
You can’t add or subtract fractions without
common denominators, so what is the
easiest way to make the two equal using
multiplication?
Multiply the 6x under 5 by 6x to get a denominator
of 36x^2
Now that both have the same least common denominator, whatever you do to
change a denominator, you must do the same to the numerator. So, you must
also multiply the 5 in the numerator by 6x to make the fractions ready to add
or subtract.
Finding the LCD by factoring the denominators
Ex: Find the least common denominator of (2x+3)/ 30x^5 and 1/8x
30x^5 = 2 • 3 • 5 • x^5
Factor the denominators
8x= 2^3 • x
2^3, 3, 5, x^5
For each of the factors, find the highest
version
2^3 • 3 • 5 • x^5
Multiply those factors
= 120x^5
So, the LCD is 120x^5
..to continue..
So, we started with (2x+3)/ 30x^5 and 1/ 8x and we found that the LCD
was 120x^5.
When we change the denominator to create the LCD, we must also multiply
the numerator with the same number that we used to multiply the
original denominator. So, for (2x+3)/30x^5 what would we multiply
30x^5 by to get 120x^5?
30x^5 • ? = 120x^5
Divide both sides by 30x^5
? = 120x^5 / 30x^5
Simplify
?=4
Now that we’ve found the number we multiplied the denominator by to get
120x^5, we must also multiply the numerator by that number, 4, to get
a final fraction of 8x+12/ 120x^5
8x • ? = 120x^5
Divide both sides by 8x
? = 120 x•x•x•x•x / 8•x
Simplify
?= 15x^4
Now also multiply the numerator by 15x^4 to get a final fraction of 15x^4/
120x^5.
Now that we have learned how to find the LCD we can add and
subtract fractions using the same methods.
Ex: 2/ 3x^3 + 3/2x^2
3 • 2 • x^3 = 6x^3
First find the LCD
Use the highest factors and multiply them to find the
LCD
To change 3x^3 to 6x^3 we multiplied by 2 so we must also multiply 2 by 2 to
get 4/6x^3
To change 2x^2 to 6x^3 we multiplied by 3x so we must also multiply 3 by 3x
to get 9x/ 6x^3.
Now we have we have common denominators, we can use our adding methods to
solve the equation
4/6x^3 + 9x/6x^3
= 4 + 9x/ 6x^3
Add the numerators together and put them over the
common denominator
Add with Unlike Binomial Denominators
A binomial denominator is a denominator with a number and a variable
added or subtracted such as x-1 or 4x+2
To find the LCD between two fractions with binomial denominators,
Ex: 3/(x-1) and 2/(4x+2), you multiply each binomial denominator by the
other binomial denominator. In other words, multiply (x-1) by (4x+2)
and (4x+2) by (x-1) to get the LCD for each.
EXAMPLE
3/(x-1) + 2/(4x+2)
First we must find the LCD
3/(x-1)(4x+2) + 2/(4x+2)(x-1)
Multiply each denominator by the other
3(4x+2)/(x-1)(4x+2) + 2(x-1)/(x-1)(4x+2)
Remember that you must multiply
the numerators by the same
values that you multiply the
denominators by.
12x+12/(x-1)(4x+2) + 2x-2/(x-1)(4x+2)
Distribute in the numerators
*hint* keep the denominators in
factored form for possible
canceling when the numerators
are added.
12x+12+2x-2/(x-1)(4x+2)
Add the numerators together
14x+10/(x-1)(4x+2)
If you have to solve an equation with variables in the numerator on one
side and variables in the denominator on the other side, you can use
cross multiplication to solve.
Ex: Solve 8/(3+x) = 4x/2
8(2) = (3+x)(4x)
16 = 12x + 4x^2
0 = 4x^2+12x-16
0 = 4 (x^2 + 3x – 4)
0 = 4 (x+4)(x-1)
x+4=0 x-1=0
x= -4 and x=1
Write the original equation
Cross multiply
Multiply out and simplify
Subtract 16 from each side
Factor out the 4
Factor out the quadratic equation
Set both binomials equal to 0
Solve each equation
In order to solve any kind of rational equation, you can multiply by the LCD
to solve the equation.
Ex: Solve 3/x + 5/7 = 6/x
Write the original equation
Find the LCD which turns out to be 7x
In order to solve for x only multiply the numerators by the LCD to get the
equation out of fraction form
7x(3)/x + 7x(5)/7 = 7x(6)/x
21x/x + 35x/7 = 42x/x
21+5x= 42
5x= 42-21
5x= 21
x= 21/5
Multiply numerators by the LCD
Multiply out
Simplify the fractions
Subtract 21 from both sides
Subtract and simplify
Divide both sides by 5 to find x
When you have to solve a rational equation with denominators that can be
factored, factor first and then multiply by the LCD
Ex: Solve 4/(x+2) + 3/ (x^2+4x+4) = 1
4/(x+2) + 3/(x+2)(x+2)
Factor the denominator of the 2nd
expression
4(x+2)^2/(x+2) + 3(x+2)^2/(x+2)^2= 1 (x+2)^2
Multiply the numerators by the
LCD, (x+2)^2, to both sides of
the equation
4(x+2)(x+2)/(x+2) + 3(x+2)^2/(x+2)^2 =1(x+2)^2 Simplify by canceling common
factors in the numerators and
denominators
4(x+2) + 3 = x^2+4x+4
Simplify and expand binomials
4x+8 +3 = x^2+4x+4
Distribute
4x+11 = x^2 +4x+4
Add like terms
0 = x^2-7
Subtract 4x+11 to set the equation to 0
7= x^2
Subtract 7 on both sides to isolate the variable
7
=x
Square root both sides of the equation to solve
for x
The End
Congratulations, you are now a
chapter 11 genius