Chapter 11Rational Expressions and

Equations

By: Nick Fanning and Rhyan Gelinas

11.1 Proportions

• Reciprocal Property of Proportionsa/b = c/d then b/a = d/c

• Cross Product Propertya/b = c/d then ad = bc

11.1 Proportions

• Ex 1) 2/3 = 4/6 then

3/2 = 6/4• Ex 2)

2/3 = 4/6 then

2*6 = 3*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 = kStep 2: Substitute

4/2 = KStep 3: Solve

k = 2

11.2 Direct and Inverse Variation

• 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: Substitute4 * 2 = K

Step 3: Solvek = 8

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

11.3 Simplifying Rational Expressions

• 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

11.4 Multiplying and Dividing Rational Expressions

• Multiplicationa/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

11.4 Multiplying and Dividing Rational Expressions

• Divisiona/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

11.5 Adding and Subtracting with Like Denominators

• Additiona/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

11.5 Adding and Subtracting with Like Denominators

• Subtractiona/c – b/c = (a-b) /c

Ex) Subtract 5/2x + x/2xStep 1: Write equation

5/2x - x/2x Step 2: Subtract

5-x/2xStep 3: Simplify

(5-x)/2x

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)/40xStep 1: Factor

12x = 2^2 * 3 * x40x^4 = 2^3 * 5 * x^4

Step 2: Find the highest power of each factor

2^3, 3, 5, x^4Step 3: Multiply

2^3 *3 *5 *x^4 = 120x^4

11.7 Rational Equations

• Rational Equation = equation that contains rational expressions

Ex) Solve 5/(y+2) = y/3Step 1: Write

5/(y+2) = y/3Step 2: Cross Multiply

5(3) = y(y+2)Step 3: Simplify

y^2 + 2y = 15Step 4: Write

y^2 +2y -15 = 0Step 5: Factor

(y+5)(y-3) = 0Answer = y = -5, 3

Links

• Ration Practice– http://www.regentsprep.org/Regents/Math/rati

o/PracRatio.htm• Direct and Inverse Variation

– http://www.mathsteacher.com.au/year10/ch17_variation/05_directvariation/25direct.htm

• Fraction Practice– http://www.aaamath.com/fra.html