Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

1.E: Algebra Fundamentals (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 1.E.1

Reduce to lowest terms.

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

1. 715

3. 3118

Exercise 1.E.2

Simplify.

  1. (12)
  2. ((58))
  3. ((a))
  4. (((a)))
Answer

1. 12

3. a

Exercise 1.E.3

Graph the solution set and give the interval notation equivalent.

  1. x10
  2. x<0
  3. 8x<0
  4. 10<x4
  5. x<3 and x1
  6. x<0 and x>1
  7. x<2 or x>6
  8. x1 or x>3
Answer

1. [10,);

Figure 1.E.1

3. [8,0);

Figure 1.E.2

5. [1,3);

Figure 1.E.3

7. R

Figure 1.E.4

Exercise 1.E.4

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)
Answer

1. x8

3. 12x32

5. x1 or x>5

7. x>4

Exercise 1.E.5

Simplify.

  1. |34|
  2. |(23)|
  3. (|4|)
  4. ((|3|))
Answer

1. 34

3. 4

Exercise 1.E.6

Determine the values represented by a.

  1. |a|=6
  2. |a|=1
  3. |a|=5
  4. |a|=a
Answer

1. a=±6

2.

Exercise 1.E.7

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
Answer

1. 15

3. 4

5. 827

7. 15

9. 6

11. 24

13. 347

15. 50

17. 14

Exercise 1.E.8

Simplify.

  1. 38
  2. 518
  3. 60
  4. 6
  5. 7516
  6. 8049
  7. 340
  8. 381
  9. 381
  10. 332
  11. 325027
  12. 31125
Answer

1. 62

3. 0

5. 534

7. 235

9. 333

11. 5323

Exercise 1.E.9

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

  1. 12
  2. 314
  3. 318
  4. 7325
  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.
Answer

1. 3.464

3. 2.621

5. 82 centimeters

Exercise 1.E.10

Multiply

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

1. 6x2+2x4

3. 3a2+15ab+6b2

Exercise 1.E.11

Combine like terms.

  1. 5x2y3xy24x2y7xy2
  2. 9x2y2+8xy+35x2y28xy2
  3. a2b27ab+6a2b2+12ab5
  4. 5m2n3mn+2mn22nm4m2n+mn2
Answer

1. x2y10xy2

3. 5ab+1

Exercise 1.E.12

Simplify.

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

1. x2+16x+3

3. a2ab+2b2

Exercise 1.E.13

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=236
  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.
Answer

1. 114

3. 80

5. 30

7. 26

9. 60π

11. $855

13. 1.6 seconds

Exercise 1.E.14

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
Answer

1. x7

3. 21x6y3z4

5. 2a75b2c

7. x20y1216z4

9. 25a4b6c10

11. 27a98b15c3

Exercise 1.E.15

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.
Answer

1. 1.333×1015

3. 7×1023

5. $128.57

Exercise 1.E.16

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
Answer

1. x22x+2

3. 2a2b22ab13

5. 12x26x+3

7. 2x23x20

9. 2x45x3+16x213x+20

11. 8a3+b3

13. 27x327x2+9x1

15. x42x2y2+y4

17. 9a+27b3

19. x23x+2

21. 5x2x+2+5x4

23. a2a+2

Exercise 1.E.17

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?
Answer

1. 53

3. 145

5. 118

7. 13

9.

11. 3

13. R

15. 72

17. s=Aπr2πr

19. 5,13

21. Length: 21 meters; Width: 12 meters

23. 42 miles per hour

Exercise 1.E.18

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<13 and 4x1>10
  18. 3x18 and 2x+523
  19. 5x3<2 or 5x3>2
  20. 13x1 or 13x1
  21. 5x+6<6 or 9x2>11
  22. 2(3x1)<16 or 3(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?
Answer

1. (,5);

Figure 1.E.5

3. [6,);

Figure 1.E.6

5. (,13);

Figure 1.E.7

7. ;

Figure 1.E.8

9. [45,);

Figure 1.E.9

11. [3,2);

Figure 1.E.10

13. (1,32);

Figure 1.E.11

15. (12,92);

Figure 1.E.12

17. (114,5);

Figure 1.E.13

19. (,15)(1,);

Figure 1.E.14

21. R;

Figure 1.E.15

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

Sample Exam

Exercise 1.E.19

Simplify.

  1. 53(12|252|)
  2. (12)2(32|34|)3
  3. 760
  4. 5332
  5. Find the diagonal of a square with sides measuring 6 centimeters.
Answer

1. 38

3. 1415

5. 62 centimeters

Exercise 1.E.20

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
Answer

2. a3c38b6

4. 2x2+x+7

6. 27a3b3

Exercise 1.E.21

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
Answer

1. 83

3. 1123

5. y=caxb

Exercise 1.E.22

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>14 or 3x7<14
Answer

2. R;

Figure 1.E.16

4. (,73)(7,);

Figure 1.E.17

Exercise 1.E.23

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?
Answer

2. Length: 31 inches; width: 36 inches

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


1.E: Algebra Fundamentals (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

  • Was this article helpful?

Support Center

How can we help?