math is cool! especially chapter 11 in the algebra 1 book! slide show by andrew sublett and jeehee...
TRANSCRIPT
Math is cool! Especially Chapter 11 in the Algebra 1
book!
Slide Show by Andrew Sublett and Jeehee Cho
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
Proportion – an equation that states that two ratios are equal
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) Factor out x^3 in the denominator
x^3/ x^3 (x + 1) Cancel out common x^3 in the numerator
and denominator
1/ (x+1) Simplify
2. (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) Multiply numerators with
numerators and denominators with denominators
Ex: (3x)/(2x^2) ÷ (3)/(4x) = (3x)/(2x^2) • (4x)/ (3) When dividing fractions, flip
the fraction you are diving by and
multiply out
(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^4Using 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
Multiplying Rational Expressions
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) 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).
= (-2x-1)/(7x+1) 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 You can’t add or subtract fractions without common denominators, so what is the easiest way to make the two equal using
multiplication?
6x • 6x = 36x^2 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 First find the LCD
3 • 2 • x^3 = 6x^3 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 Add the numerators together and put them over the common denominator
= 4 + 9x/ 6x^3
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.
EXAMPLE3/(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 Write the original equation8(2) = (3+x)(4x) Cross multiply16 = 12x + 4x^2 Multiply out and simplify0 = 4x^2+12x-16 Subtract 16 from each side0 = 4 (x^2 + 3x – 4) Factor out the 40 = 4 (x+4)(x-1) Factor out the quadratic equationx+4=0 x-1=0 Set both binomials equal to 0x= -4 and x=1 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 7xIn 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 Multiply numerators by the LCD21x/x + 35x/7 = 42x/x Multiply out 21+5x= 42 Simplify the fractions5x= 42-21 Subtract 21 from both sides5x= 21 Subtract and simplifyx= 21/5 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) = 14/(x+2) + 3/(x+2)(x+2) Factor the denominator of the 2nd
expression4(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 binomials4x+8 +3 = x^2+4x+4 Distribute 4x+11 = x^2 +4x+4 Add like terms0 = x^2-7 Subtract 4x+11 to set the equation to 07= x^2 Subtract 7 on both sides to isolate the
variable 7 = x Square root both sides of the equation to
solve for x
Ex: Solve 4/(x+2) + 3/ (x^2+4x+4) = 14/(x+2) + 3/(x+2)(x+2) Factor the denominator of the 2nd
expression4(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 binomials4x+8 +3 = x^2+4x+4 Distribute 4x+11 = x^2 +4x+4 Add like terms0 = x^2-7 Subtract 4x+11 to set the equation to 07= x^2 Subtract 7 on both sides to isolate the
variable 7 = x Square root both sides of the equation to
solve for x