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1.9 Review Exercises and Sample Exam

Review Exercises

    Review of Real Numbers and Absolute Value

      Reduce to lowest terms.

    1. 56120
    2. 5460
    3. 15590
    4. 315120

      Simplify.

    1. (12)

    2. ((58))

    3. ((a))

    4. (((a)))

      Graph the solution set and give the interval notation equivalent.

    1. x10

    2. x<0

    3. 8x<0

    4. 10<x4

    5. x<3andx1

    6. x<0andx>1

    7. x<2orx>6

    8. x1orx>3

      Determine the inequality that corresponds to the set expressed using interval notation.

    1. [8,)

    2. (,7)

    3. [12,32]

    4. [10,0)

    5. (,1](5,)

    6. (,10)(5,)

    7. (4,)

    8. (,0)

      Simplify.

    1. |34|

    2. |(23)|

    3. (|4|)

    4. ((|3|))

      Determine the values represented by a.

    1. |a|=6

    2. |a|=1

    3. |a|=5

    4. |a|=a

    Operations with Real Numbers

      Perform the operations.

    1. 1415+320

    2. 23(34)512

    3. 53(67)÷(514)

    4. (89)÷1627(215)

    5. (23)3

    6. (34)2

    7. (7)282

    8. 42+(4)3

    9. 108((35)22)
    10. 4+5(3(23)2)
    11. 32(7(4+2)3)
    12. (4+1)2(36)3
    13. 103(2)332(4)2
    14. 6[(5)2(3)2]46(2)2
    15. 73|6(32)2|
    16. 62+5|32(2)2|
    17. 12|62(4)2|3|4|
    18. (52|3|)3|4(3)2|32

    Square and Cube Roots of Real Numbers

      Simplify.

    1. 38

    2. 518

    3. 60

    4. 6

    5. 7516

    6. 8049

    7. 403

    8. 813

    9. 813

    10. 323

    11. 250273
    12. 11253

      Use a calculator to approximate the following to the nearest thousandth.

    1. 12

    2. 314

    3. 183

    4. 7253

    5. Find the length of the diagonal of a square with sides measuring 8 centimeters.

    6. Find the length of the diagonal of a rectangle with sides measuring 6 centimeters and 12 centimeters.

    Algebraic Expressions and Formulas

      Multiply.

    1. 23(9x2+3x6)
    2. 5(15y235y+12)
    3. (a25ab2b2)(3)
    4. (2m23mn+n2)6

      Combine like terms.

    1. 5x2y3xy24x2y7xy2

    2. 9x2y2+8xy+35x2y28xy2

    3. a2b27ab+6a2b2+12ab5

    4. 5m2n3mn+2mn22nm4m2n+mn2

      Simplify.

    1. 5x2+4x3(2x24x1)
    2. (6x2y2+3xy1)(7x2y23xy+2)
    3. a2b2(2a2+ab3b2)
    4. m2+mn6(m23n2)

      Evaluate.

    1. x23x+1 where x=12

    2. x2x1 where x=23

    3. a4b4 where a=3 and b=1

    4. a23ab+5b2 where a=4 and b=2

    5. (2x+1)(x3) where x=3

    6. (3x+1)(x+5) where x=5

    7. b24ac where a=2, b=4, and c=1

    8. b24ac where a=3, b=6, and c=2

    9. πr2h where r=23 and h=5

    10. 43πr3 where r=263

    11. What is the simple interest earned on a 4 year investment of $4,500 at an annual interest rate of 434%?

    12. James traveled at an average speed of 48 miles per hour for 214 hours. How far did he travel?

    13. The period of a pendulum T in seconds is given by the formula T=2πL32 where L represents its length in feet. Approximate the period of a pendulum with length 2 feet. Round off to the nearest tenth of a foot.

    14. The average distance d, in miles, a person can see an object is given by the formula d=6h2 where h represents the person’s height above the ground, measured in feet. What average distance can a person see an object from a height of 10 feet? Round off to the nearest tenth of a mile.

    Rules of Exponents and Scientific Notation

      Multiply.

    1. x10x2x5
    2. x6(x2)4x3
    3. 7x2yz33x4y2z
    4. 3a2b3c(4a2bc4)2
    5. 10a5b0c425a2b2c3
    6. 12x6y2z36x3y4z6
    7. (2x5y3z)4
    8. (3x6y3z0)3
    9. (5a2b3c5)2
    10. (3m55n2)3
    11. (2a2b3c3ab2c0)3
    12. (6a3b3c2a7b0c4)2

      Perform the operations.

    1. (4.3×1022)(3.1×108)

    2. (6.8×1033)(1.6×107)

    3. 1.4×10322×1010

    4. 1.15×10262.3×107

    5. The value of a new tablet computer in dollars can be estimated using the formula v=450(t+1)1 where t represents the number of years after it is purchased. Use the formula to estimate the value of the tablet computer 212 years after it was purchased.

    6. The speed of light is approximately 6.7×108 miles per hour. Express this speed in miles per minute and determine the distance light travels in 4 minutes.

    Polynomials and Their Operations

      Simplify.

    1. (x2+3x5)(2x2+5x7)

    2. (6x23x+5)+(9x2+3x4)

    3. (a2b2ab+6)(ab+9)+(a2b210)

    4. (x22y2)(x2+3xyy2)(3xy+y2)

    5. 34(16x2+8x4)
    6. 6(43x232x+56)
    7. (2x+5)(x4)

    8. (3x2)(x25x+2)

    9. (x22x+5)(2x2x+4)

    10. (a2+b2)(a2b2)

    11. (2a+b)(4a22ab+b2)

    12. (2x3)2

    13. (3x1)3

    14. (2x+3)4

    15. (x2y2)2

    16. (x2y2+1)2

    17. 27a2b9ab+81ab23ab
    18. 125x3y325x2y2+5xy25xy2
    19. 2x37x2+7x22x1
    20. 12x3+5x27x34x+3
    21. 5x321x2+6x3x4
    22. x4+x33x2+10x1x+3
    23. a4a3+4a22a+4a2+2
    24. 8a410a22

    Solving Linear Equations

      Solve.

    1. 6x8=2

    2. 12x5=3

    3. 54x3=12

    4. 56x14=32

    5. 9x+23=56

    6. 3x810=52

    7. 3a52a=4a6

    8. 85y+2=47y

    9. 5x68x=13x

    10. 176x10=5x+711x

    11. 5(3x+3)(10x4)=4

    12. 62(3x1)=4(13x)

    13. 93(2x+3)+6x=0

    14. 5(x+2)(45x)=1

    15. 59(6y+27)=213(2y+3)
    16. 445(3a+10)=110(42a)
    17. Solve for s: A=πr2+πrs

    18. Solve for x: y=mx+b

    19. A larger integer is 3 more than twice another. If their sum divided by 2 is 9, find the integers.

    20. The sum of three consecutive odd integers is 171. Find the integers.

    21. The length of a rectangle is 3 meters less than twice its width. If the perimeter measures 66 meters, find the length and width.

    22. How long will it take $500 to earn $124 in simple interest earning 6.2% annual interest?

    23. It took Sally 312 hours to drive the 147 miles home from her grandmother’s house. What was her average speed?

    24. Jeannine invested her bonus of $8,300 in two accounts. One account earned 312% simple interest and the other earned 434% simple interest. If her total interest for one year was $341.75, how much did she invest in each account?

    Solving Linear Inequalities with One Variable

      Solve. Graph all solutions on a number line and provide the corresponding interval notation.

    1. 5x7<18

    2. 2x1>2

    3. 9x3

    4. 37x10

    5. 613(x+3)>13

    6. 73(2x1)6

    7. 13(9x+15)12(6x1)<0

    8. 23(12x1)+14(132x)<0

    9. 20+4(2a3)12a+2

    10. 13(2x+32)14x<12(112x)
    11. 43x+5<11

    12. 5<2x+1513

    13. 1<4(x+1)1<9

    14. 03(2x3)+110

    15. 1<2x54<1
    16. 23x3<1
    17. 2x+3<13and4x1>10

    18. 3x18and2x+523

    19. 5x3<2or5x3>2

    20. 13x1or13x1

    21. 5x+6<6or9x2>11

    22. 2(3x1)<16or3(12x)<15

    23. Jerry scored 90, 85, 92, and 76 on the first four algebra exams. What must he score on the fifth exam so that his average is at least 80?

    24. If 6 degrees less than 3 times an angle is between 90 degrees and 180 degrees, then what are the bounds of the original angle?

Answers

  1. 715
  2. 3118
  3. 12
  4. a

  5. [10,);

  6. [8,0);

  7. [1,3);

  8. ;

  9. x8

  10. 12x32

  11. x1orx>5

  12. x>4

  13. 34
  14. 4

  15. a=±6

  16. Ø

  1. 15

  2. −4

  3. 827

  4. −15

  5. −6

  6. −24

  7. 347

  8. −50

  9. 14

  1. 62

  2. 0

  3. 534
  4. 253
  5. 333
  6. 5233
  7. 3.464

  8. 2.621

  9. 82 centimeters

  1. 6x2+2x4

  2. 3a2+15ab+6b2

  3. x2y10xy2

  4. 5ab+1

  5. x2+16x+3

  6. a2ab+2b2

  7. 114

  8. 80

  9. 30

  10. 26

  11. 60π

  12. $855

  13. 1.6 seconds

  1. x7

  2. 21x6y3z4
  3. 2a75b2c
  4. x20y1216z4
  5. 25a4b6c10
  6. 27a98b15c3
  7. 1.333×1015

  8. 7×1023

  9. $128.57

  1. x22x+2

  2. 2a2b22ab13

  3. 12x26x+3

  4. 2x23x20

  5. 2x45x3+16x213x+20

  6. 8a3+b3

  7. 27x327x2+9x1

  8. x42x2y2+y4

  9. 9a+27b3

  10. x23x+2

  11. 5x2x+2+5x4
  12. a2a+2

  1. 53
  2. 145
  3. 118
  4. 13

  5. Ø

  6. −3

  7. 72

  8. s=Aπr2πr

  9. 5, 13

  10. Length: 21 meters; Width: 12 meters

  11. 42 miles per hour

  1. (,5);

  2. [6,);

  3. (,13);

  4. Ø;

  5. [45,);

  6. [3,2);

  7. (1,32);

  8. (12,92);

  9. (114,5);

  10. (,15)(1,);

  11. ;

  12. Jerry must score at least 57 on the fifth exam.

Sample Exam

      Simplify.

    1. 53(12|252|)
    2. (12)2(32|34|)3
    3. 760

    4. 5323
    5. Find the diagonal of a square with sides measuring 6 centimeters.

      Simplify.

    1. 5x2yz1(3x3y2z)

    2. (2a4b2ca3b0c2)3
    3. 2(3a2b2+2ab1)a2b2+2ab1

    4. (x26x+9)(3x27x+2)

    5. (2x3)3

    6. (3ab)(9a2+3ab+b2)
    7. 6x417x3+16x218x+132x3

      Solve.

    1. 45x215=2

    2. 34(8x12)12(2x10)=16

    3. 125(3x1)=2(4x+3)

    4. 12(12x2)+5=4(32x8)

    5. Solve for y: ax+by=c

      Solve. Graph the solutions on a number line and give the corresponding interval notation.

    1. 2(3x5)(7x3)0

    2. 2(4x1)4(5+2x)<10

    3. 614(2x8)<4

    4. 3x7>14or3x7<14

      Use algebra to solve the following.

    1. Degrees Fahrenheit F is given by the formula F=95C+32 where C represents degrees Celsius. What is the Fahrenheit equivalent to 35° Celsius?

    2. The length of a rectangle is 5 inches less than its width. If the perimeter is 134 inches, find the length and width of the rectangle.

    3. Melanie invested 4,500 in two separate accounts. She invested part in a CD that earned 3.2% simple interest and the rest in a savings account that earned 2.8% simple interest. If the total simple interest for one year was $138.80, how much did she invest in each account?

    4. A rental car costs $45.00 per day plus $0.48 per mile driven. If the total cost of a one-day rental is to be at most $105, how many miles can be driven?

Answers

  1. 38

  2. 1415

  3. 62 centimeters

  4. a3c38b6
  5. 2x2+x+7

  6. 27a3b3

  7. 83
  8. 1123

  9. y=caxb

  10. ;

  11. (,73)(7,);

  12. Length: 31 inches; width: 36 inches

  13. The car can be driven at most 125 miles.