1. Chapter 11 Rational Expressions and Equations
By: Nick Fanning and Rhyan Gelinas

2. 11.1 Proportions Reciprocal Property of Proportions
a/b = c/d then b/a = d/c
Cross Product Property
a/b = c/d then ad = bc

3. 11.1 Proportions
Ex 1)
2/3 = 4/6 then
3/2 = 6/4
Ex 2)
2*6 = 3*4

4. 11.2 Direct and Inverse Variation
Direct Variation
The variables x and y vary directly if and only if for a constant k
y/x = k or y =kx where k ≠ 0
Ex)
Find an equation that relations x and y such that x and y vary directly and y = 4 and x = 2
Step 1: Write direct variation model
y/x = k
Step 2: Substitute
4/2 = K
Step 3: Solve
k = 2

5. 11.2 Direct and Inverse Variation
The variable x and y vary inversely if for a constant k
xy = k or y = k/x where k ≠ 0
Ex)
Find an equation that relations x and y such that x and y vary inversly and y = 4 and x = 2
Step 1: Write inverse variation model
xy = k
Step 2: Substitute
4 * 2 = K
Step 3: Solve
k = 8

6. 11.3 Simplifying Rational Expressions
Rational Number = number that can be written as the quotient of 2 integers
Rational Expression = a number whose number and denominator are a nonzero polynomial

7. 11.3 Simplifying Rational Expressions
ac/ bc = a * c / b * c = a/b
Ex) Simplify 14x/7
Step 1: Write equation
14x/7
Step 2: Rationalize
2 * 7 * x / 7
Step 3: Simplify 2x

8. 11.4 Multiplying and Dividing Rational Expressions
Multiplication
a/b * c/d = ac/bd
Ex) Multiply 4x/2 * 2x/4
Step 1: Write equation
4x/2 * 2x/4
Step 2: Multiply
8x/8
Step 3: Simplify x

9. 11.4 Multiplying and Dividing Rational Expressions
Division
a/b ÷ c/d = a/b * d/c
Ex) Divide 4x/2 ÷ 2x/4
Step 1: Write equation
4x/2 ÷ 2x/4
Step 2: Multiply
4x/2 * 4/2x = 16x/ 4x
Step 3: Simplify 4

10. 11.5 Adding and Subtracting with Like Denominators
Addition
a/c + b/c = (a+b) /c
Ex) Add 5/2x + x-5/2x
Step 1: Write equation
5/2x + x-5/2x
Step 2: Add
x/2x
Step 3: Simplify
1/2

11. 11.5 Adding and Subtracting with Like Denominators
Subtraction
a/c – b/c = (a-b) /c
Ex) Subtract 5/2x + x/2x
Step 1: Write equation
5/2x - x/2x
Step 2: Subtract
5-x/2x
Step 3: Simplify
(5-x)/2x

12. 11.6 Adding and Subtracting unlike Denominators
LCD = Least Common Denominator
Use to add and subtract unlike denominators
Ex) Find the LCD of 1/12x and (2+x)/40x
Step 1: Factor
12x = 2^2 * 3 * x
40x^4 = 2^3 * 5 * x^4
Step 2: Find the highest power of each factor
2^3, 3, 5, x^4
Step 3: Multiply
2^3 *3 *5 *x^4 = 120x^4

13. 11.7 Rational Equations
Rational Equation = equation that contains rational expressions
Ex) Solve 5/(y+2) = y/3
Step 1: Write
5/(y+2) = y/3
Step 2: Cross Multiply
5(3) = y(y+2)
Step 3: Simplify
y^2 + 2y = 15
Step 4: Write
y^2 +2y -15 = 0
Step 5: Factor
(y+5)(y-3) = 0
Answer = y = -5, 3

14. Links Ration Practice Direct and Inverse Variation Fraction Practice
Direct and Inverse Variation
Fraction Practice