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Final Exam Review HOMEWORK: Review all in class review problems, cumulative review problems below, ALL exams, quizzes and homework problems, ALL Story Problem Applications from sec 1.5, 2.4, 3.4 & 5.6, Solving equations algebraically (linear, quadratic, rational, radical, literal, polynomial, linear systems of equations etc) and terms and definitions. Cum. Review Cum. Review Cum. Review Cum Review Cum. Review Cum Review Cum Review 619 - 620 1, 2, 9, 11, 13 – 17, 20, 22 – 25 578 – 579 1 – 9, 12 - 16, 18, 19, 21 – 28, 30 513 – 514 1 – 7, 10 – 20, 22 – 26, 28 422 – 423 1 – 5, 7, 8, 10, 12 – 19 358 – 359 1 – 11, 13 – 17, 19 – 25 245 1 – 15 203 - 204 1 – 20 Page 1 Final Exam Review From Exam 4: Quadratics 1. Solve 2 x 4 x 7 algebraically. Give solution in exact simplifies form. Show all your work. 2 Must first get 0 on one side 2x 4x 7 0 2 a =2, b = -4, c = -7 (4) (4) 4 (2)(7) x 22 2 4 16 56 x 22 Must put in ( ) around b! Watch negative signs! (-#)^2 is always positive! x 4 72 22 x 4 36 2 22 x 46 2 22 Simplify under radical Must reduce! Factor and cancel like factors! Page 2 Final Exam Review x 2 2 3 2 22 Factor out the 2. Then cancel. x 23 2 2 Note: You CANNOT cancel the 2. You can NEVER cancel over addition and subtraction! Not equal to x 3 2 ! 2. Solve algebraically. Show all your work. x 1 5 x x 1 5 x x 1 5 x 2 Must isolate radical first! 2 ( x 1)( x 1) 5 x x 2x 1 5 x 2 Square both sides. NEVER square each term! *** MUST FOIL!!! ***** x 3x 4 0 Get 0 on one side ( x 4)( x 1) 0 Factor! 2 x40 or x 1 0 Use zero property to set each factor=0 Page 3 Final Exam Review x = -4 or x = 1 Solve MANDATORY Check in ORIGINAL EQUATION!!! x 1 5 x x = -4 4 1 5 (4) 4 1 9 4 1 3 x=1 1 1 5 (1) 1 1 4 1 1 2 11 So answer is only x=1 Page 4 Final Exam Review A model rocket is launched straight upwards from the top of a 100 ft balcony. The initial velocity is 160 ft/s. The height of the rocket in feet h(t) at any time t in seconds is given by h(t ) 16t 160t 100 Find the following. 3. 2 a) Find the coordinates of the vertex of h(t ) 16t 160t 100 . Show your work! b 160 5 x-coordinate = 2a 2(16) b y-coordinate = h( ) h(5) 16 5 160 5 100 500 2a 2 2 ANSWER 5, 500 b) How long does it take the rocket to reach its maximum height? Include units. ANSWER 5 sec c) What is the maximum height the rocket will reach? Include units. ANSWER 500 ft d) How long does it take for the rocket to hit the ground? Include units. Show your work! Hits ground when h = 0 0 16t 160t 100 2 Solve with QE Page 5 Final Exam Review 160 160 4 (16) 100 t 2 16 2 ANSWER 10.59 sec (Round answer to 2 decimal places) 4. Rationalize the denominator and simplify the expression, if possible. Show all your work. 5 7 3 1 5. Solve 4 x 7 3 2 x or x 3 2 x 6 algebraically. 2 Show all work. Give final answer in interval notation. 4x 7 3 2x 1 x 3 2x 6 2 2 x 10 x 6 4 x 12 x5 6 3x 2 x Page 6 Final Exam Review MUST graph on number line (important part of work) 1 4 x 7 3 2 x or x 3 2 x 6 2 Solve 4 x 7 3 2 x and Solve 4 x 7 3 2 x or 1 x 3 2x 6 2 1 x 3 2x 6 2 Solve 4 x 7 3 2 x and 1 x 3 2x 6 2 Page 7 Final Exam Review Simplify completely. Assume all variables represent positive real numbers. Do not leave negative exponents in your answer. Show all your work. 8x y 27 x z 8 1 8x y 27 x z 2y 3x z 8 1 3 4 5 1 4 3 53 3 4 5 1 3 1 3 1 3 8 (y ) (2 ) y 8y (3 ) x z 27 x z 27 ( x ) ( z ) 2y z 3x 3 4 9 5 1 1 3 3 3 9 1 4 3 1 3 1 4 3 3 13 3 5 1 3 3 13 9 13 5 13 5 3 1 4 3 Note: WE NEVER convert to radical form unless instructed to do so. When the problem is given in exponential form, keep it in this form! ALWAYS SHOW ALL STEPS! EVERY STEP ABOVE is required! Multiply and simplify. Show all your work. (3 10 2 )(4 5 5 8 ) MUST FOIL!!! Reduce ALL radicals! Page 8 Final Exam Review Combine. Assume that all variables represent positive numbers. Show all your work. 6 50a b 4ab 18ab ab 3 5 3 2 8ab Always reduce all radicals to like terms first. Then combine with dist property. 6 5 2a ab b 4ab 3 2ab b ab 2 2 6 5ab 30ab 2 2 4 2 2 2 2 2ab 2 2ab 4ab 3b 2ab ab 2 2ab 2 2ab 12ab 2 2ab 2ab 2 2ab 30 12 2ab 2ab 2 20ab 2 2ab Write the expression by using exponents rather than radical 3 notation. x 12 5 Which of the following is equivalent to 5 3 ? 2 NEVER square each term. ALWAYS F.O.I.L!!! Page 9 Final Exam Review Rationalize the denominator. 2 5x NEVER square an expression. ALWAYS multiply by “1” to make denominator be powers of the index. Solve 2( x 5) 48 algebraically. Give the answer exactly. 2 Isolate the squared term first. Never FORGET ! Do not multiply all out! Page 10 Final Exam Review Equations vs expressions: NEVER DROP = in EQUATIONS!!!!! Linear: Solve 2 3(4 x (5 2 x) 1) 2 (3x 1) 2 Simplify 2 3(4 x (5 2 x) 1) Page 11 Final Exam Review Polynomial: Solve. Show all your work. 12n 3n 3 Always factor completely! Watch constants! Factor 12n 3n 3 Note: There is NO = sign, so answer is an expression! DO NOT add an equal sign when one is not present! Page 12 Final Exam Review Solve 14 x 2 algebraically 2 x 5x 3 2 x 1 x 3 2 Note: When there is an equation (= sign) ALWAYS multiply by LCD!!! MUST CHECK ANSWER! Page 13 Final Exam Review Simplify 14 x 2 2 x 5x 3 2 x 1 x 3 2 Note: When there is NO = sign, NEVER NEVER NEVER multiply by LCD!!! You MUST put each term over the LCD by multiplying by “1” to make each term have a denominator equal to the LCD and them combine! NEVER DROP DENOMINATORS! Rewrite entire fraction in EACH STEP when you simplify! Page 14 Final Exam Review Literal equations: Solve 3b 7au 4 y 1 for u. 5 x 2u Strategy: Clear Fractions by mult by LCD. Get all terms with the variable you want on one side and terms without that variable on the other. Factor out the variable you want. Then divide by the coefficient. Page 15 Final Exam Review 6 x xy y 6 xa 10 xb 3ay 5by Simplify 50 x 18x 15ax 25bx 9ax 15bx 2 2 3 Answer: 2 2 3x y 2(5 x 3) Page 16 Final Exam Review Solve 3x 2 y 11 2 x 5 y 18 Answer: (-1,4) Solve 2 x 5 y 18 5 x y9 2 5 Answer: Infinite solution x x y 9 2 Page 17 Final Exam Review Domain: The set of all input values: Find the domain of the following: Give answer in interval notation if possible. 5x 2 f ( x) 16x 36 x 3 Answer: (, 3 ) ( 3 ,0) (0, 3 ) ( 3 , ) 2 2 2 2 g ( x) 3 5 x Answer: (, 3 ] 5 7,3,2,4,5, 1,8,9 Answer: 7, 2, 5, 8 y x Answer: 3, 6 Page 18 Final Exam Review Lines: Parallel Lines have same slopes. Perpendicular lines have opposite reciprocal slopes. Slope = rise/run Find the equation of the line perpendicular to 2 x 5 y 7 going through (-1, 3). Give answer in slope intercept form. Find the equation of the line parallel 2 x 3 going through (4, -1). Give answer in slope intercept form. Find the equation of the line perpendicular 5 y 1 going through (-2, -3). Give answer in slope intercept form. Page 19 Final Exam Review Find the equation of the line going through the points (-3,5) and (2, -1). Find the x and y intercepts of the line going through the points (-3, 5) and (2, -1). Sketch the graph. Page 20 Final Exam Review Algebraic Properties A) Associative Property of Addition B) Associative Property of Multiplication C) Commutative Property of Addition D) Commutative Property of Multiplication E) Identity Property of Addition F) Identity Property of Multiplication G) Inverse Property of Addition H) Inverse Property of Multiplication I) Distributive Property J) Multiplication Property of 0 i) iii) v) 3 3 0 5 5 _____ 5 (6 7) (5 6) 7 3 5 1 5 3 vii) ii) ____ ___ 2 6 2 4 4 ( ) 3 7 5 7 15 ___ 3 3 0 5 5 ____ iv) 5 (6 7) 5 (7 6) vi) 3 3 1 5 5 viii) ____ _____ 2(3 5) 2(5 3) ____ Page 21 Final Exam Review Classify the numbers by set. Check all boxes that the number belongs to Number Natural Whole Integers Rational Irrational Real 16 0 -5.36 2 4 4 16 2 5 3 5 8 3 8 2.345345345 2 5 17 List the rational #’s in the 1st column above: List the irrational #’s in the 1st column above Page 22 Final Exam Review Applications: Linear: Example: The sum of 2 consecutive odd integers is -56. Find the integers. Answer: -27 & -29 Example: Suppose you plan to borrow $6000 from 2 lenders to pay for your tuition next year. One lender charges 10% simple interest and the other charges 4% simple interest. How much did you borrow from each lender if you paid a total of $391.83 in interest after 1 year? Page 23 Final Exam Review Answer: $2530.50 from the 10% lender and $3469.50 from the 4% lender. Example: Suppose you invested $5000 into 2 accounts, a CD paying 6% simple interest and a stock paying 8% simple interest. If you receive $706.40 in interest after 2 years, how much was invested in each account? a) Set up as a an equation in one variable and solve b) Set it up as a system of equations and solve. Answer: a) Let x = amt in CD. Then 0.06x*2 + (5000 – x) * .08*2 = 706.4 b) Let x = amt in CD and y = amt in stock x + y = 5000 0.06x*2 + 0.08y*2 = 706.4 or .12x + .16y = 706.4 CD: $2340; Stock: $2660 Page 24 Final Exam Review Example: How many liters of 40% antifreeze should be added to 4 L of a 10% antifreeze solution to produce a 35% antifreeze solution? a) Set up as a an equation in one variable and solve b) Set it up as a system of equations and solve. Answer: a) Equation: Let x = amt of 40% antifreeze soln 0.40x + 0.10*4 = 0.35 ( 4+x) b)Equation: Let x = amt of 40% antifreeze soln, and y be the amt of 35% soln x+4=y 0.40x + 0.10*4 = 0.35*y Answer: 20 L of 40% solution Page 25 Final Exam Review Example: How many liters of 2.5% bleach should be added to a 10% bleach solution to produce 600 ml a 5% bleach solution? a) Set up as a an equation in one variable and solve b) Set it up as a system of equations and solve. Answer: a) Equation: Let x = amt of 2.5% bleach soln 0.025x + (600-x)0.10 = 0.05 *600 b)Equation: Let x = amt of of 2.5% bleach soln and y be the amt of 10% soln x + y = 600 0.025x + 0.10y = 0.05*600 Answer: 400 ml of 2.5% and 200 ml of 10% Page 26 Final Exam Review Example: A favorite blend of coffee is a mixture of Columbian costing $5.10 per pound and Hazelnut costing $6.40 per pound. How much of each should be used to produce 10 lbs of the blend costing $5.85/lb? a) Set up as a an equation in one variable and solve b) Set it up as a system of equations and solve. Answer: a) Equation: Let x = amt of Columbian coffee 5.10x + 6.4(10-x) = 5.85*10 b)Equation: Let x = amt of Columbian coffee and y be the amt of Hazelnut coffee x + y = 10 5.10x + 6.4y = 0.1*5.85 Answer: 4.23 lbs Columbian and 5.77 lbs Hazelnut Page 27 Final Exam Review Example: Two trains are 190mi apart and travel toward each other on the same road. They meet in 2 hours. One travels 4 mph faster than the other. What is the average speed of each train? Set up as an equation in one variable and solve. Answer: D=rt table. TOTAL DISTANCE (SUM) Let x be the rate of the first train. Equation 2x +2(x+4)=190 1st train 45.5 mph, 2nd train 49.5 mph Page 28 Final Exam Review Example: Bob can ride his bike 2 mph faster than Johann can rollerblade. If it takes Bob 3.1 hours to ride his bike down a trail and Johann 4.3 to rollerblade the same trail, how fast were each of them going? Answer: D=rt table. Distance same Let x be the rate of Johann. Equation 3.1(x+2)=4.3x Johann 3 mph, Bob 5 mph 1 6 1 6 Page 29 Final Exam Review Example: It takes a plane 3 hours to travel 600 miles with a tailwind and 5 hours to return the same distance with a headwind. Find the speed of the plane and the speed of the wind. Answer: D=rt table. {headwind/tailwind with same distance} Let x be the rate of plane in still air and w = rate of wind. Equation 3(x+w)=600 5(x-w)=600 Plane 160 mph, wind 40 mph Page 30 Final Exam Review Example: A motorist travels 80 mi while driving in a bad rainstorm. In sunny weather, the motorist drives 20mph faster and covers 120 mi in the same amount of time. Find the speed of the motorist in the rainstorm and in sunny weather. Answer: D=rt table. SAME TIME Let x be the rate in rainstorm 80 120 Equation x x 20 In rainstorm 40 mph, Sunny Day 60 mph Page 31 Final Exam Review Example: Brooke walks 2 km/hr slower than her older sister Pam. If Broke can walk 12 km in the same amount of time that Pam can walk 18 km, find their speeds. Answer: D=rt table. SAME time t=D/r Let x be the rate of Pam 12 18 Equation x2 x Pam 6 mph, Brooke 4 mph Page 32 Final Exam Review Example: A bicyclist rides 60 mi against the wind and returns 60 mi with the wind. His average speed for the return trip is 2 mph faster. How fast did the cyclist ride each way if the total time for the trip was 11 hrs? Answer: D=rt table. Total Time Let x be the rate of against wind 60 60 Equation 11 x x2 10 mph against the wind, and 12 mph with the wind Page 33 Final Exam Review Example: A local theater charges $8.50 for student tickets and $12 for general admission. If they sold 167 tickets for $1864, how many of each type did they sell? Equation: x y 167 8.50x 12 y 1864 Answer: 40 student tickets and 127 general admission. Example: The width of a rectangle is 5 in less than 3 times the length. The area is 28 in2. Find the dimensions. Equation: 28 = l(3l-5) Answer: l=4 in, w=7in Page 34 Final Exam Review Example: A shadow cast by a yardstick is 2 ft long. A shadow cast by a tree is 11 ft long. How tall is the tree? 3 h 2 11 Answer: 16.5 ft Equation Example: A 64 oz bottle of laundry detergent costs $4.00, how much would a 100-oz bottle cost? 64 oz 100 oz 4$ x$ Answer: $6.25 Equation: Page 35 Final Exam Review Example: A chemist mixes water and alcohol in a 7 to 8 ratio. If she makes a 450 ml solution, how much is water and how much is alcohol. 7 x 7 x or 8 450 x 15 450 Answer: 210 ml water and 240 ml alcohol. Equation: Example: The cost of a monthly texting plan is $12 per month plus $0.10 per text. a) Find a formula for the cost C as a function of the number of text messages x. b) How much will it cost you to send/receive 50 text messages? 2500 text messages? c) How many text messages would you need to send/receive to make it more economical to switch over to the unlimited plan with a fixed rate of $65 per month? Answer: a) C 12 0.1x b) $17, $262 c) > 530 Page 36