Extra Problems: Unit 0

Transcription

1 Extra Problems: Unit 0 These are example problems, mostly from tests and quizzes that I have given in the past. Make sure you understand and can do all assigned textbook problems, activities, problems from handouts, and quizzes in addition to these problems. 1.) Determine which quadrant each of the following points lie in. If none, say so. () (-2, 3) () (3, -2) () (-3, -3) () (0, 3) (e.) (1, 10) (f.) (-2, 1000) 2.) Write each of the following sets in interval notation. {x 7 < x 10} {x x > -2} 3.) True or False: Write true if the statement is always true for the indicated conditions. Write false otherwise. ( & ) If x > 0 and y < 0: If 4.) Simplify the following: (Do not have absolute values in your answer.) Assume x < 0, 5x 3 3x 2 Assume x > 3, 3 - x 5.) Rewrite the following sentence as an inequality involving an absolute value. The distance between -1 and x is at most 2. 6.) Evaluate x 2 y 3 + xy 2 y x 2 where x = 2 and y = -1. (I recommend that you do this one without a calculator.) 7.) Simplify the following completely. Write all your answers in exponential form. Do not have negative exponents in your answer. Cancel everything possible. No variable should appear more than once in your answer. Assume all variables are positive. Get rid of all parentheses.

2 8.) Rewrite the following expressions as one radical with everything possible pulled out. Rationalize all denominators. Assume all variables are positive. 9.)* Simplify the following radicals assuming x and y could be any real numbers. We have not discussed when you need absolute value around your answer, but this is also a review topi Feel free to ask! 10.) Simplify the following expressions. Write your answer in regular polynomial form. 11.) Factor and simplify the following expressions completely.

3 12.) Perform the indicated operations and simplify. Do not have complex fractions in your answer. Write your answer as a single fraction and reduce it completely. Rationalize the denominator where appropriate. 13.) Simplify the following expressions completely. Do not have negative exponents in your answer. 14.) Simplify. Then rewrite your answer using radical notation. 15.) Write 20,300 in scientific notation. 16.) Subtract 2x 2 3x + 2 from x 2 + 2x 1. Simplify your answer. 17.) Multiply (2a + b) by (a 2 3b). Simplify your answer.

4 18.) Solve each of the following for x algebraically and then confirm your answer graphically. 7x 2 1 = 0 (x + 1) 3 = x x 3 + 3x 2 2x 3 = 0 3x 3 7x 2 + 3x = 0 e. (x + 2)(x - 5) = ) Fill in the blanks. Note = the set of complex numbers, = the set of rational numbers, = the set of real numbers, = the set of integers, = the set of natural numbers, and = the set of irrational numbers. Put each of the following sets in order as indicated: and : Fill in the blank with an example: : 20.) Name the mathematical property indicated: ) k(x + y) = kx + ky ) x + y = y + x ) a(bc) = (ab)c 21. Solve the following equations for x: ) ) ) ) e.) 22.) If a racewalker takes 1760 steps per mile and walks 5 miles per hour for 3 consecutive hours, how many steps did he take? Express your answer in scientific form. 23.) Calculate the following: 4 is 5 percent of what?

5 What is 5 percent of 4? 5 is what percent of 4? 24. Solve the following application problems. A laboratory keeps two acid solutions on han One is 20 percent acid and the other is 35 percent aci An order is received for 25 liters of a 26 percent acid solution. How much 20 percent acid solution and how much 35 percent acid solution should be used to fill this order? Two semicircles are placed at opposite ends of a square as shown. Find the side length of the square if the total area enclosed is 100 square units. Recall that the area of a circle is p r 2 where r is the radius. Suppose your four highest test scores are: 65, 72, 68, 79 out of 100 each. If the final counts as two tests what percent do you need to get on the final to get a 2.5 or above in the course? (A 77% average is neede) A farmer plans to use 180 feet of fencing to enclose a rectangular region, using part of a straight river bank instead of fencing as one side of the rectangle. Find the area of the region if the length of the side parallel to the river bank is one and a half times the length of an adjacent side. e. A runner leaves her house to run on her favorite 6 mile out and back course (3 miles out and 3 miles back) at a rate of 8 minutes per mile. Her husband heads out on the same course 12 minutes later at a rate of 10 minutes per mile. How far from their home will they pass each other? 25.) Solve the following inequalities. Write your answer in simplified interval notation. 3x + 1 > 12 or -2x + 2 > 5 3x + 1 < 12 and -2x x x 4 and (x < 2 or x 4) 26.) Simplify the following. Put your answer in the form a + bi where a and b are reals. 6i(5 + 3i)

6 i 291 e. (-i) 651 f. -i ) Find all complex solutions to the following equations using any method you wish. Show all written work. ) ) ) ) 4x 4-15x 2-4 = 0 28.) Solve for x exactly: ) 8x x 3 = 8 ) 2x 2 + 5x 1 = 0 29.) Suppose you have a 6 ft. by 8 ft. built-in swimming pool. You want to put in a rectangular concrete border all the way aroun You can afford 72 square feet worth of concrete. How wide is your border? 30.) Solve the following inequalities algebraically and sketch a graph of the solution. Write your answers in interval notation. 31.) Solve for all real values of x: 2x = 3

7 e. 32.) Suppose a hose can fill a pool by itself in 12 hours. A smaller hose takes 15 hours to fill the pool. Suppose you are filling the pool with the larger hose for 3 hours when you add the smaller hose. How long will it take to finish filling the pool? 33.) A right triangle has an area of 40 ft 2 and a hypotenuse that is 2 feet longer than one of its sides. Let x denote the length of that side. Find the length of its legs. (Recall for a right triangle a 2 + b 2 = c 2 where a & b are the legs and c is the hypotenuse. Also the area is ½ base height.) equation used to solve this problem: solution: Last Update: Oct. 3, 2017 Answers: 1. () II () IV () III () none (e.) I (f.) II 2. ) (7, 10] ) (-2, ) 3. ) false ) true ) true 4. ) -5x 3 ) 3x 2 ) x (-1) 3 + 2(-1) = = ) ) ) 8. ) ) 9. ) ) x + 3 ) 10. ) 6x 3-11x 2-18x + 20 ) 8x x 2 y + 6xy 2 + y 3 ) -4xy - 3y 2 z 11. ) (3 - x)(1 - yz) ) (4x - y) 2 ) 2(x - 2)(x 2 + 2x + 4) 12. ) ) -1 ) )

8 (x 2 + 2x 1) (2x 2 3x +2) = -x 2 + 5x (2a + b)(a 2 3b) = 2a 3 6ab + a 2 b 3b ± (7)/7 0, -1-3/2, -1, 1 0, e. 0, (2) (answers will vary) distributive property commutative property of addition associative property of multiplication 21.) ) ) ) ) e.) 22.) ) ) 80, ) 0.2, ) ) ) 10 liters at 35% and 15 liters at 20% ) x = 7.48 units ) 89% or above ) ft 2 e.) 2 miles 25.) ) (-, -3/2) (11/3, ), ) [-3/2, 11/3), ) [2, 4] ) [1, 2) {4} 26.) Simplify the following. Put your answer in the form a + bi where a and b are reals i

9 13i e. i f. -1, so since the remainder is 3:i 291 = i 3 = -i 27.) ) x = 9 ; ) no solution ) x = ±8 ) x = ±i/2, ±2 28.) ) x = ½ or -2 ) 29.) 2 feet 30.) (-, 4/15] [4/3, ) [-1, 5) 31.) 0, no real solns e ) 5 hours 33.) 4x 3 + 4x = 0 or x 3 + x = 0, so x = The legs are feet and 80/ = feet. You can be sure that you got the only possible value for x by checking to make sure your xmax is an upper bound of the zeros. X can't be negative in this problem, so any negative number will do for xmin.