Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

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1 Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order: 1 st : powers, roots, logrithms nd : multipliction, division 3 rd : ddition, subtrction Exmple: = 35 (Incorrect) 8 1 = 8 5 = 75 (Correct) (This is importnt to keep in mind for clcultor computtions.) Grouping symbols [ ] nd ( ) re used to further specify the order of opertions. An expression within prentheses is to be evluted before continuing to opertions outside the prentheses (following the rithmetic hierrcy within prentheses, s bove). B. Fundmentl Properties of Rel Numbers Definition: The set of ll rtionl nd irrtionl numbers is clled the set of rel numbers. Rtionl Numbers: rtio numbers cn be expressed s rtio of two integers Ex: 7, ½, 1.39, 43/8, b, 444. Irrtionl Numbers: number whose deciml form does not terminte nd does not hve repeting pttern. 3 Ex:, π, 5, e Let, b, c be rel numbers. Then: Additive Multiplictive Commuttive Property: + b = b b = Associtive Property: ( + b) + c = + (b + c) (b)c = (bc) Distributive (b + c) b c (These rules llow us to simplify nd evlute complex expressions.) Pre-Session Review 1

2 Ex: ( + b)(c + d) =? = ( +b)c + ( + b)d (Distributive property) = c + bc + d + bd (Distributive property gin) We cn lso go in reverse: x + bx = ( + b)x (Clled fctoring ) Adding (subtrcting) like terms: In n expression, like terms cn be dded (subtrcted) by dding (subtrcting) the coefficients. Ex: 1x + x = (1 + )x = 3x (5x - 4y - 17) + (y - 3x -) = (5-3)x + (-4 - )y + (-17 - ) = x - y - 19 Properties of zero nd one (Cncelltion) Addition: + (-) = (1-1) = = 6x - 6x = (6-6)x = x = 1 Multipliction: = 1 = 1 x x + + y y = 1 ( 3x + )( b) ( b + )( 3x + ) = 3x 3x + + b = 1 b + b = b + b b + C. Solving nd Grphing Equtions Equtions in one vrible: Ex: x - = 6 x - = x = 8 Rule: to mintin equlities, lwys perform the sme opertion to both sides of the eqution. Ex: x + b = c (Solve for x) x + b = c -b -b x = c - b Solve: 1. Subtrct b from both sides x = c b x = c b. Divide both sides by Pre-Session Review

3 Prctice: solve for x: 3(x + ) =! (x + ) 1. Multiply out. Isolte terms involving x on left hnd side, other terms on right hnd side 3. Fctor 4. Divide both sides by (3 + ) Equtions in two vribles: Ex: y = 1 x No unique solution here, for ech x we get different y A grph shows the reltionship between x nd y: x y = 1/x / -1/4-1/8 1/8 1/4 1/ 1 3-1/3-1/ undefined / 1/3 Y y = 1/x X Pre-Session Review 3

4 Ex: y = x - 9 (prbol) x y = x Pre-Session Review 4

5 B. Liner Equtions nd Functions Ex: y = x + 5 x y = x Y 1 y = x X The Generl Liner Form: y = mx + b y = dependent vrible x = independent vrible m = slope coefficient b = intercept (constnt term) Definition: A liner eqution in two vribles contins only terms tht re constnts or constnts times one vrible to the first power. Exmples: x = 4 y = ¾x + 1 z = -x!. 3x + 4y = 5 ( equivlent to Y = -3/4 x + 5/4 ) Not Liner: y = x +1 x + 4xy = 8 x + y = Pre-Session Review 5

6 Definition: Slope mesures the chnge in the dependent vrible per one-unit chnge in the independent vrible. Exmple: y =!3.4 x + 8 Slope: For ech 1-unit increse in x we hve 3.4-unit decrese in y, since slope =!3.4 Recll: slope cn be thought of s rtio rise over run, or the verticl distnce divided by the horizontl distnce between ny two points on the line. Intercept: if x =, then y = 8, so the line crosses the y-xis t y = 8. y = - 3.4x Y X Problem:. Wht is the slope of the line given by the eqution 4x + y = 5? b. Grph the line. 5 Y X Pre-Session Review 6

7 Applictions with liner equtions: 1. A vendor finds tht she sells units/dy when the price is $6/unit, nd 16 units/dy when the price is $8. If the demnd curve is liner nd price is the dependent vrible,. wht is its slope? b. wht is the eqution? P 4 Demnd Q. Brekeven Anlysis Suppose selling price is fixed t $8 (so the totl revenue function is TR = 8Q ) Suppose the totl cost function is TC = Q Find the Brekeven quntity. Brekeven Y TR = TC 6 so 8Q = Q.... Solve for Q 5 4 $ Q Pre-Session Review 7

8 Liner Inequlities An inequlity specifies rnge for vrible, not just unique vlue. Exmples: x $ 1 z < 3-1 # y # 6 Grphiclly: y $ x + 3 The line y = x + 3 is the boundry All points bove nd to the right stisfy the inequlity y $ x + 3 Systems of liner inequlities: Exmple: Show grphiclly ll points tht stisfy both inequlities. y $ x - 4 b. y # 3x - 4 Pre-Session Review 8

9 Appliction of systems of liner inequlities: Liner Progrmming Exmple: Inputs K nd L re used to produce output Q K costs $3 per unit L costs $9 per unit The firm s input budget is $18. Mterils bought must be loded into the compny truck, with cpcity of 4, pounds. K weighs 1 pounds per unit L weighs pounds per unit Show the combintions of K nd L tht re fesible given these two constrints. 3K + 9L # 18 K # -3L + 6 1K + L # 4 K # -L + 4 K $ K $ L $ L $ Pre-Session Review 9