1. Math is cool! Especially Chapter 11 in the Algebra 1 book! Slide Show by Andrew Sublett and Jeehee Cho

2. 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

3. 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

4. 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

5. 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/xSet up the proportions with the given information 6(x) = 12(5)Multiply diagonals using the cross product property 6x = 60 Multiply out x = 10 pencilsDivide each side by 6 *hint: don’t forget your units!

6. 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

7. *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

8. 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

9. 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

10. (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 Multiplying Rational Expressions

11. 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/12Simplify fraction x/2

12. 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

13. 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^2You 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^2Multiply 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.

14. 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^5Factor the denominators 8x= 2^3 x 2^3, 3, 5, x^5For each of the factors, find the highest version 2^3 3 5 x^5 Multiply those factors = 120x^5So, the LCD is 120x^5

15. ..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^5Divide 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 xxxxx / 8xSimplify ?= 15x^4 Now also multiply the numerator by 15x^4 to get a final fraction of 15x^4/ 120x^5.

16. 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^3Use 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

17. 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.

18. 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)

19. 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 equation 8(2) = (3+x)(4x)Cross multiply 16 = 12x + 4x^2Multiply out and simplify 0 = 4x^2+12x-16Subtract 16 from each side 0 = 4 (x^2 + 3x – 4)Factor out the 4 0 = 4 (x+4)(x-1)Factor out the quadratic equation x+4=0 x-1=0Set both binomials equal to 0 x= -4 and x=1Solve each equation

20. 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/xWrite 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)/xMultiply numerators by the LCD 21x/x + 35x/7 = 42x/xMultiply out 21+5x= 42Simplify the fractions 5x= 42-21Subtract 21 from both sides 5x= 21Subtract and simplify x= 21/5Divide both sides by 5 to find x

21. 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 2 nd expression 4(x+2)^2/(x+2) + 3(x+2)^2/(x+2)^2= 1 (x+2)^2Multiply 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)^2Simplify by canceling common factors in the numerators and denominators 4(x+2) + 3 = x^2+4x+4Simplify and expand binomials 4x+8 +3 = x^2+4x+4Distribute 4x+11 = x^2 +4x+4Add like terms 0 = x^2-7Subtract 4x+11 to set the equation to 0 7= x^2Subtract 7 on both sides to isolate the variable 7 = xSquare root both sides of the equation to solve for x Ex: Solve 4/(x+2) + 3/ (x^2+4x+4) = 1 4/(x+2) + 3/(x+2)(x+2)Factor the denominator of the 2 nd expression 4(x+2)^2/(x+2) + 3(x+2)^2/(x+2)^2= 1 (x+2)^2Multiply 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)^2Simplify by canceling common factors in the numerators and denominators 4(x+2) + 3 = x^2+4x+4Simplify and expand binomials 4x+8 +3 = x^2+4x+4Distribute 4x+11 = x^2 +4x+4Add like terms 0 = x^2-7Subtract 4x+11 to set the equation to 0 7= x^2Subtract 7 on both sides to isolate the variable 7 = xSquare root both sides of the equation to solve for x

22. The End Congratulations, you are now a chapter 11 genius