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

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

11.1 Proportions Ex 1) 2/3 = 4/6 then 3/2 = 6/4 Ex 2) 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 = k Step 2: Substitute 4/2 = K Step 3: Solve k = 2

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

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 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 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

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

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

11.5 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

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

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

Links Ration Practice Direct and Inverse Variation Fraction Practice http://www.regentsprep.org/Regents/Math/ratio/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