Math 060/ Final Exam Review Guide/ 2010-2011/ College of the Canyons General Information: The final exam is a 2-hour timed exam. There will be approximately 40 questions. There will be no calculators or notes allowed. You will be given the formulas for how to factor the sum and difference of cubes. a 3 b3 (a b)(a 2 ab b 2 ) and a 3 b3 (a b)(a 2 ab b 2 ) Topics: Solve linear inequalities. (Section 2.8) Write and graph equations of lines, including finding slope. (Sections 3.2 -3.6) Solve systems of linear equations. (Sections 4.1-4.3) Multiply and divide polynomials. (Sections 5.2-5.5) Simplify expressions with zero and negative exponents. (Section 5.4) Factor polynomials. (Sections 6.1-6.5) Add, subtract, multiply, divide rational expressions. (Sections 7.2, 7.3, 7.5, 7.6) Solve equations. (Sections 2.3, 2.4, 6.6, 7.7) Solve word problems. (Sections 2.6, 2.7, 4.4, 4.5, 6.7, 7.8) Study Tip: Write one math question (from the sample final, your past exams, and the book) per 3x5 card. On the back of the 3x5 card write where you found the problem and the answer. Mix the cards up when you practice solving the problems. Write notes to yourself on the back of the cards if you need to remember formulas or other steps. Formulas to Remember: Slope between two points: m Slope of parallel lines: m2 y2 x2 y1 x1 m1 Slope of perpendicular lines: m1 m2 1 or m2 Slope-Intercept Form of a Line: y mx b Point-Slope Form of a Line: y y1 m( x x1 ) 1 m1 An x-intercept is ( x, 0). In other words, let y = 0. A y-intercept is (0, y). In other words, let x = 0. Vertical lines have the equation x = a with an undefined slope. Horizontal lines have the equation y = b with a slope of zero. The graph of the lines y mx and Ax By 0 go through the origin. Formulas to Remember, continued: Exponent Rules (See page 348.) Product am an am n Power (a m )n a m n Quotient am a m n if a 0 n a Product to a Power (ab) n a nb n n Quotient to a Power a an if b 0 b bn Zero Exponent a0 1 if a 0 Negative Exponent 1 if a 0 a n an 1 a n Quotient to a Negative Power a b a n if a 0 n b a n if a 0 , b 0 Polynomials: Special Forms ( A B)( A B) A2 B 2 ( A B) 2 A2 2 AB B 2 ( A B) 2 A2 2 AB B 2 Uniform Motion: (Rate)(Time) = Distance OR (Distance) / (Rate) = Time If wind or current affects the rate, use x y for traveling “with” and use x y for traveling “against.” The object’s speed is represented by x, and the wind (or current) speed is represented by y. Simple Interest: (Principal$)(Rate%) = Interest$ [This formula is good over a length of time = 1 year.] [Remember to change % into decimal form.] Mixture: (Quantity)(Concentration%) = Amount [Remember to change % into decimal form.] Work: 1 individual time 1 individual time 1 where t represents time together. t Complementary angles add up to 90 . Supplementary angles add up to 180 . Area of a rectangle: A LW Pythagorean Theorem: a2 b2 Area of a triangle: A c2 1 bh 2 bh 2 Sample Problems for the Math 060 Final Exam, 2010-2011 (Sullivan, Struve, Mazzarella combo book) 1. Solve the inequalities. Graph the solution set on a number line and write the solution set in interval notation. a. 1 2x (2.8 #71) 3 1 b. (2.8 #77) ( x 4) x 8 2 c. 4( x 1) (2.8 #79) 3( x 1) x d. 4(2w 1) (2.8 #85) 3(w 2) 5(w 2) 2. Graph each linear equation by finding its intercepts. Write the coordinates of the x-intercept: ( , ). Write the coordinates of the y-intercept: ( , ). a. 3x 6 y 18 (3.2 #67) b. (3.2 #69) x 5 y 15 c. 9x 2 y 0 (3.2 #73) 3. Graph the following lines. a. x 5 b. y (3.2 #83) (3.2 #85) 6 4. Graph the line that contains the given point and has the given slope. a. (2, 3); m 0 b. (2,1); m (3.3 #51) 2 3 c. ( 1, 4); m (3.3 #53) 5 3 d. (0,0); m is undefined (3.3 #55) (3.3 #57) 5. Use the slope and y-intercept to graph each line. a. y b. y 5 x 2 c. 3x 2 y 10 d. y (3.4 #39) x 3 x 3 2 (3.4 #45) (3.4 #51) (3.4 #81) 6. Write the equation of the line. Your answer should be in slope-intercept form y mx b . a. Through the points ( 3, 2) and (1, 4) (3.5 #41) b. Through the points ( 3, 11) and (2, 1) (3.5 #43) c. Through the points ( 2,3) and (4, 6) (3.5 #67) d. Through the point (10, 2) and parallel to the line 3x 2 y 5 (3.6 #51) e. Through the point ( 1, 10) and parallel to the line x 2 y f. Through the point ( 4, 1) and perpendicular to the line y g. Through the point (7,5) and perpendicular to the y-axis. (3.6 #53) 4 4x 1 (3.6 #57) (3.6 #61) 7. Solve the following systems of linear equations. If there is one solution, write the solution as an ordered pair (x,y). Otherwise if there are infinitely many solutions or no solution, state so. a. Solve by graphing: 3x y 1 6x 2 y 4 b. Solve by substitution or elimination: c. Solve by substitution or elimination: d. Solve by substitution or elimination: (4.1 #31) x y 2x y 3x y 1 6x 2 y 2 2x 4 y 0 5x 2 y 6 8. Find the product. a. ( x 2 y )3 ( 2 xy 3 ) (5.2 #85) b. ( x 2 3)( x 2 1) (5.3 #55) c. ( x 2) 2 (5.3 #73) d. (5k 3) 2 (5.3 #75) e. ( x 2 y ) 2 (5.3 #77) f. ( x 2)( x 2 3x 1) 7 2 (5.3 #83) (4.2 #17) (4.3 #21) (4.3 #29) 9. Simplify. Write answers with positive exponents. (If you have a number raised to a positive exponent, find its value.) a. (24ab)0 b. 2 2 5 (5.4 #69) 3 2n m2 c. (5.4 #55) d. 1 42 e. 5 2m f. 21 y 2 z 3y 2z (5.4 #73) (5.4 #75) (5.4 #79) 3 3 1 (5.4 #87) 10. Divide and simplify. a. 4 x2 2 x 2x (5.5 #13) b. 9a3 27a 2 3 3a 2 (5.5 #15) c. x4 x3 10 x 4 x 2 (5.5 #37) d. 2 x3 7 x 2 10 x 5 2x 1 (5.5 #47) e. x3 2 x 2 8 x2 2 (5.5 #53) 11. Factor completely. If the polynomial cannot be factored, say that it is prime. (In Section 6.3 the author asks students to factor by trial and error or by grouping. This will not be the case on the final exam. You may choose a method that works for you.) a. x3 x2 (6.5 #47) x 1 b. x2 100 (6.5 #19) c. 9 a2 (6.5 #33) d. 12x3 2x2 2x (6.5 #71) e. 8 x 3 27 y 3 f. (6.4 #51) x2 10x 25 (6.4 #29) g. 16x2 24x 9 (6.4 #33) h. 4z 2 12z 9 (6.4 #37) i. 27 x3 (6.4 #49) j. 9x 2 (6.4 #85) y2 k. 2 x2 5x 3 l. (6.3 #23) 4 p 2 11 p 3 (6.3 #29) m. 6n2 17n 10 (6.3 #35) n. 5w2 13w 6 (6.3 #49) o. 6x2 17 x 12 (6.3 #85) p. 30x 22x2 24x3 (6.3 #93) q. 6 x 2 ( x 2 1) 25 x ( x 2 1) 14( x 2 1) (6.3 #95) r. m2 9m 18 (6.2 #23) s. z 2 12z 45 (6.2 #33) t. x 2 5 xy 6 y 2 (6.2 #35) u. 2 y2 8 y 8 (6.2 #51) v. 4 x 3 32 x 2 x4 (6.2 #53) w. g 2 4 g 21 (6.2 #71) x. 15a2b4 60ab3 45a3b2 (6.1 #63) y. x 2 ( x 1) (6.1 #67) z. y 2 ( x 1) 2t 3 t 2 4t 2 (6.1 #77) 12. Perform the indicated operation. a. 3y y y 6 b. p2 1 2p 3 c. 9 x2 x2 5x 4 d. 2c 4 8 2 c 2 e. 4y 8 9 y2 (7.2 #37) 2 p2 p 6 p2 3 p 2 (7.2 #19) 2 a x2 2x 3 x2 4 x (7.2 #47) a 3a 10 2 (7.2 #59) a 2 2 3a 10 f. x2 x 2x g. 2x x y2 h. 6 2a 6 i. 3x 1 x x j. 2n 1 n2 4 3n 6 n2 n (7.5 #67) 3 m (7.5 #77) k. l. 2x2 x 2x (7.3 #31) 2y y x2 2 4a 1 a 3 9 2 m 2 3 2 5 6 2 2 9x m 10 m2 4 1 4 1 2 1 (7.5 #41) (7.5 #61) b b 4 1 b 4 2 x n. (7.3 #55) (7.6 #25) b2 2 m. b 16 b 2 b 16 1 (7.3 #35) 3 x2 9 x2 (7.6 #43) (7.6 #45) 13. Solve the equations. Check your solution. a. a 4 a 3 1 2 b. p 0.05 p 157.5 c. 0.02(2c 24) check: a 4 check: p 0.05 p 157.5 0.4(c 1) a 3 1 2 check: 0.02(2c 24) 14. Solve the equations. a. Solve for r: (2.4 #53) A P Pr t 1 h ( B b) 2 b. Solve for b: A c. Solve for y: d. Solve for S: x z 3 y 1 R 1 S (2.4 #55) (7.7 #53) 1 T (7.7 #55) 15. Solve the equations. a. n2 9n 14 0 (6.6 #37) b. 4x2 2x 0 (6.6 #39) c. n2 n 6 (6.6 #47) d. 2x3 2x2 12x 0 e. y3 3 y 2 4 y 12 f. 2a (a 1) a2 8 (6.6 #59) 0 (6.6 #61) (6.6 #73) (2.3 #35) (2.3 #51) 0.4(c 1) (2.3 #59) 16. Solve the equations. Remember to check for values of the variable which make the expressions in each rational equation undefined. a. b. c. 2 x 2 2 7 2 (7.7 #21) x 2 3 a 1 a 1 x 3 x 2 x 8 6 a 2 (7.7 #25) 1 (7.7 #35) Warning: The bold printed words shown below at the beginning of the word problems will not always appear with the word problems listed on the final. 17. Bad Investment. (2.6 #43) After Mrs. Fisher lost 9% of her investment, she had $22,750. What was Mrs. Fisher’s original investment? Variable and what it represents: _______________________________ Equation:_____________________________ Original Investment:______________ 18. Commission. (2.6 #53) Melanie receives a 3% commission on every house she sells. If she received a commission of $8571, what was the value of the house she sold? Variable and what it represents: _______________________________ Equation:_____________________________ Value of House Sold:______________ 19. Angles. (2.7 #13) Find two supplementary angles such that the measure of the first angle is 10 less than three times the measure of the second. Variable and what it represents: _______________________________ Equation:_____________________________ Measures of both angles:______________ 20. Uniform Motion. (2.7 #39) Two boats leave a port at the same time, one going north and the other traveling south. The north-bound boat travels 16 mph faster than the south-bound boat. If the southbound boat is traveling at 7 mph, how long will it be before they are 1430 miles apart? (You may use the table below to help set up your equation and solve the problem.) = Equation: __________________________ Length of time: __________________ 21. Uniform Motion. (2.7 #43) A 360-mile trip began on a freeway in a car traveling at 62 mph. Once the road became a 2-lane highway, the car slowed to 54 mph. If the total trip took 6 hours, find the time spent on each type of road. (You may use the table below to help set up your equation and solve the problem.) = Equation:____________________________ Time spent on freeway: ________ Time spent on 2-lane highway:_______ 22. Uniform Motion. (2.7 #45) Carol knows that when she jogs along her neighborhood greenway, she can complete the route in 10 minutes. It takes 30 minutes to cover the same distance when she walks. If her jogging rate is 4 mph faster than her walking rate, find the speed at which she jogs. (You may use the table below to help set up your equation and solve the problem.) = Equation:____________________________ Jogging speed:_______________ 23. Uniform Motion. (4.4 #37) Vanessa and Richie are riding their bikes down a trail to the next campground. Vanessa rides at 10 mph while Richie rides at 7 mph. Since Vanessa is a little speedier, she stays behind and cleans up camp for 30 minutes before leaving. How long has Richie been riding when Vanessa is 7 miles ahead of Richie? (You may use the table below to help set up your equation and solve the problem.) = Equation:____________________________ Length of time:________ 24. Uniform Motion. (4.4 #33) Suppose that Jose bikes into the wind for 60 miles and it takes him 6 hours. After a long rest, he returns (with the wind at his back) in 5 hours. Determine the speed at which Jose can ride his bike in still air and determine the effect that the wind had on his speed. (You may use the table below to help set up your system of equations and solve the problem.) = System of Equations:________________________ Jose’s speed:________ Wind speed:______ 25. Uniform Motion. (4.4 #39) With a tailwind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the effect of the wind and the average airspeed of the Piper. (You may use the table below to help set up your system of equations and solve the problem.) = System of Equations:__________________________ Piper’s speed:_____ Wind speed:____ 26. Oakland Baseball. (4.4 #19) The attendance at the games on two successive nights of Oakland A’s baseball was 44,000. The attendance on Thursday’s game was 7000 more than two-thirds of the attendance at Friday night’s game. How many people attended the baseball game each night? Variables and what they represent: _______________________________ System of Equations:__________________________ Attendance on Thursday:_______ Attendance on Friday:_________ 27. Angles. (4.4 #27) The measure of one angle is 15 more than half the measure of its complement. Find the measures of the two angles. Variables and what they represent: _______________________________ System of Equations:__________________________ Measure of one angle: ____________ Measure of the other angle:_________ 28. Interest. (4.5 #29) Harry has $10,000 to invest. He invests in two different accounts, one expected to return 5% and the other expected to return 8%. If he wants to earn $575 for the year, how much should he invest at each rate? (You may use the table below if you wish.) = System of Equations or Equation:__________________________ Amount invested at 5%:______________ Amount invested at 8%:______________ 29. Mixture. (4.5 #39) A lab technician needs 60 ml of a 50% saline solution. How many ml of 30% saline solution should she add to a 60% saline solution to obtain the required mixture? (You may use the table below if you wish.) = System of Equations or Equation:__________________________ Quantity of 30% solution:______________ 30. Mixture. (4.5 #41) How many liters of 10% silver must be added to 70 liters of 50% silver to make an alloy that is 30% silver? You may use the table below if you wish.) = System of Equations or Equation:__________________ Quantity of 10% solution: ___________ 31. Area of a Triangle. (6.7 #29) The sail on a sailboat is in the shape of a triangle. If the height of the sail is 3 times the length of the base and the area is 54 square feet, find the dimensions of the sail. Equation:____________________ Base of sail:____________ Height:_____________ 32. Pythagorean Theorem. (6.7 #31) Your big-screen TV measures 50 inches on the diagonal. If the front of the TV measures 40 inches across the bottom, find the height of the TV. Equation:____________________ Height of TV:_____________ 33. Area of a Rectangle. (6.7 #33) The length of a rectangle is 1 mm more than twice the width. If the area is 300 square mm, find the dimensions of the rectangle. Equation:__________________________________ Length:_________ Width:________ 34. Uniform Motion. (7.8 #75) While training for an iron man competition, Tony bikes for 60 miles and runs for 15 miles. If his biking speed is 8 times his running speed and it takes 5 hours to complete the training, how long did he spend on his bike? = Equation:__________________________________ Length of time: ________________ 35. Uniform Motion. (7.8 #79) You have a 20-mile commute into work. Since you leave very early, the trip going to work is easier than the trip home. You can travel to work in the same time that it takes for you to make it 16 miles on the trip back home. Your average speed coming home is 7 miles per hour slower than your average speed going to work. What is your average speed going to work? = Equation:__________________________________ Average speed to work: _____________ 36. Uniform Motion. (7.8 #73) A boat can travel 12 km down the river in the same time it can go 4 km up the river. If the current in the river is 2 km per hour, how fast can the boat travel in still water? = Equation: _______________________ Boat’s speed: ____________ 37. Work. (7.8 #65) After hitting practice for the Long Beach State volleyball team, Dyanne can retrieve all of the balls in the gym in 8 minutes. It takes Makini 6 minutes to retrieve all the balls. If they work together, to the nearest tenth of a minute, how long will it take these two player s to return the volleyballs and be ready to start the next round of hitting practice? Equation: _______________________ Time together:________________ 38. Work. (7.8 #69) It takes an apprentice twice as long as the experienced plumber to replace the pipes under an old house. If it takes them 5 hours when they work together, how long would it take the apprentice alone? Equation: _______________________ Time for apprentice alone:________________ 39. Work. (7.8 #71) Using a single hose, Janet can fill a pool in 6 hours. The same pool can be drained in 8 hours by opening a drainpipe. If Janet forgets to close the drainpipe, how long would it take her to till the pool? Equation: _______________________ Time to fill pool:________________