1.E: Algebra Fundamentals (Exercises)
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Exercise 1.E.1
Reduce to lowest terms.
- 56120
- 5460
- 15590
- 315120
- Answer
-
1. 715
3. 3118
Exercise 1.E.2
Simplify.
- −(−12)
- −(−(−58))
- −(−(−a))
- −(−(−(−a)))
- Answer
-
1. 12
3. −a
Exercise 1.E.3
Graph the solution set and give the interval notation equivalent.
- x≥−10
- x<0
- −8≤x<0
- −10<x≤4
- x<3 and x≥−1
- x<0 and x>1
- x<−2 or x>−6
- x≤−1 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.
- [−8,∞)
- (−∞,−7)
- [12,32]
- [−10,0)
- (−∞,1]∪(5,∞)
- (−∞,−10)∪(−5,∞)
- (−4,∞)
- (−∞,0)
- Answer
-
1. x≥−8
3. 12≤x≤32
5. x≤1 or x>5
7. x>−4
Exercise 1.E.5
Simplify.
- −|−34|
- −|−(−23)|
- −(−|−4|)
- −(−(−|−3|))
- Answer
-
1. −34
3. 4
Exercise 1.E.6
Determine the values represented by a.
- |a|=6
- |a|=1
- |a|=−5
- |a|=a
- Answer
-
1. a=±6
2. ∅
Exercise 1.E.7
Perform the operations.
- 14−15+320
- 23−(−34)−512
- 53(−67)÷(514)
- (−89)÷1627(215)
- (−23)3
- (−34)2
- (−7)2−82
- −42+(−4)3
- 10−8((3−5)2−2)
- 4+5(3−(2−3)2)
- −32−(7−(−4+2)3)
- (−4+1)2−(3−6)3
- 10−3(−2)332−(−4)2
- 6[(−5)2−(−3)2]4−6(−2)2
- 7−3|6−(−3−2)2|
- −62+5|3−2(−2)2|
- 12−|6−2(−4)2|3−|−4|
- −(5−2|−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.
- 3√8
- 5√18
- 6√0
- √−6
- √7516
- √8049
- 3√40
- 3√81
- 3√−81
- 3√−32
- 3√25027
- 3√1125
- Answer
-
1. 6√2
3. 0
5. 5√34
7. 23√5
9. −33√3
11. 53√23
Exercise 1.E.9
Use a calculator to approximate the following to the nearest thousandth.
- √12
- 3√14
- 3√18
- 73√25
- Find the length of the diagonal of a square with sides measuring 8 centimeters.
- 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. 8√2 centimeters
Exercise 1.E.10
Multiply
- 23(9x2+3x−6)
- −5(15y2−35y+12)
- (a2−5ab−2b2)(−3)
- (2m2−3mn+n2)⋅6
- Answer
-
1. 6x2+2x−4
3. −3a2+15ab+6b2
Exercise 1.E.11
Combine like terms.
- 5x2y−3xy2−4x2y−7xy2
- 9x2y2+8xy+3−5x2y2−8xy−2
- a2b2−7ab+6−a2b2+12ab−5
- 5m2n−3mn+2mn2−2nm−4m2n+mn2
- Answer
-
1. x2y−10xy2
3. 5ab+1
Exercise 1.E.12
Simplify.
- 5x2+4x−3(2x2−4x−1)
- (6x2y2+3xy−1)−(7x2y2−3xy+2)
- a2−b2−(2a2+ab−3b2)
- m2+mn−6(m2−3n2)
- Answer
-
1. −x2+16x+3
3. −a2−ab+2b2
Exercise 1.E.13
Evaluate.
- x2−3x+1 where x=−12
- x2−x−1 where x=−23
- a4−b4 where a=−3 and b=−1
- a2−3ab+5b2 where a=4 and b=−2
- (2x+1)(x−3) where x=−3
- (3x+1)(x+5) where x=−5
- √b2−4ac where a=2,b=−4, and c=−1
- √b2−4ac where a=3,b=−6, and c=−2
- πr2h where r=2√3 and h=5
- 43πr3 where r=23√6
- What is the simple interest earned on a 4 year investment of $4,500 at an annual interest rate of 434%?
- James traveled at an average speed of 48 miles per hour for 214 hours. How far did he travel?
- The period of a pendulum T in seconds is given by the formula T=2π√L32 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.
- 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. 2√6
9. 60π
11. $855
13. 1.6 seconds
Exercise 1.E.14
Multiply.
- x10⋅x2x5
- x6(x2)4x3
- −7x2yz3⋅3x4y2z
- 3a2b3c(−4a2bc4)2
- −10a5b0c−425a−2b2c−3
- −12x−6y−2z36x−3y4z6
- (−2x−5y−3z)−4
- (3x6y−3z0)−3
- (−5a2b3c5)2
- (−3m55n2)3
- (−2a−2b3c3ab−2c0)−3
- (6a3b−3c2a7b0c−4)−2
- Answer
-
1. x7
3. −21x6y3z4
5. −2a75b2c
7. x20y1216z4
9. 25a4b6c10
11. −27a98b15c3
Exercise 1.E.15
Perform the operations.
- (4.3×1022)(3.1×10−8)
- (6.8×10−33)(1.6×107)
- 1.4×10−322×10−10
- 1.15×10262.3×10−7
- 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.
- 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. 7×10−23
5. $128.57
Exercise 1.E.16
Simplify.
- (x2+3x−5)−(2x2+5x−7)
- (6x2−3x+5)+(9x2+3x−4)
- (a2b2−ab+6)−(ab+9)+(a2b2−10)
- (x2−2y2)−(x2+3xy−y2)−(3xy+y2)
- −34(16x2+8x−4)
- 6(43x2−32x+56)
- (2x+5)(x−4)
- (3x−2)(x2−5x+2)
- (x2−2x+5)(2x2−x+4)
- (a2+b2)(a2−b2)
- (2a+b)(4a2−2ab+b2)
- (2x−3)2
- (3x−1)3
- (2x+3)4
- (x2−y2)2
- (x2y2+1)2
- 27a2b−9ab+81ab23ab
- 125x3y3−25x2y2+5xy25xy2
- 2x3−7x2+7x−22x−1
- 12x3+5x2−7x−34x+3
- 5x3−21x2+6x−3x−4
- x4+x3−3x2+10x−1x+3
- a4−a3+4a2−2a+4a2+2
- 8a4−10a2−2
- Answer
-
1. −x2−2x+2
3. 2a2b2−2ab−13
5. −12x2−6x+3
7. 2x2−3x−20
9. 2x4−5x3+16x2−13x+20
11. 8a3+b3
13. 27x3−27x2+9x−1
15. x4−2x2y2+y4
17. 9a+27b−3
19. x2−3x+2
21. 5x2−x+2+5x−4
23. a2−a+2
Exercise 1.E.17
Solve.
- 6x−8=2
- 12x−5=3
- 54x−3=12
- 56x−14=32
- 9x+23=56
- 3x−810=52
- 3a−5−2a=4a−6
- 8−5y+2=4−7y
- 5x−6−8x=1−3x
- 17−6x−10=5x+7−11x
- 5(3x+3)−(10x−4)=4
- 6−2(3x−1)=−4(1−3x)
- 9−3(2x+3)+6x=0
- −5(x+2)−(4−5x)=1
- 59(6y+27)=2−13(2y+3)
- 4−45(3a+10)=110(4−2a)
- Solve for s:A=πr2+πrs
- Solve for x:y=mx+b
- A larger integer is 3 more than twice another. If their sum divided by 2 is 9, find the integers.
- The sum of three consecutive odd integers is 171. Find the integers.
- The length of a rectangle is 3 meters less than twice its width. If the perimeter measures 66 meters, find the length and width.
- How long will it take $500 to earn $124 in simple interest earning 6.2% annual interest?
- It took Sally 312 hours to drive the 147 miles home from her grandmother’s house. What was her average speed?
- 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=A−πr2πr
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.
- 5x−7<18
- 2x−1>2
- 9−x≤3
- 3−7x≥10
- 61−3(x+3)>13
- 7−3(2x−1)≥6
- 13(9x+15)−12(6x−1)<0
- 23(12x−1)+14(1−32x)<0
- 20+4(2a−3)≥12a+2
- 13(2x+32)−14x<12(1−12x)
- −4≤3x+5<11
- 5<2x+15≤13
- −1<4(x+1)−1<9
- 0≤3(2x−3)+1≤10
- −1<2x−54<1
- −2≤3−x3<1
- 2x+3<13 and 4x−1>10
- 3x−1≤8 and 2x+5≥23
- 5x−3<−2 or 5x−3>2
- 1−3x≤−1 or 1−3x≥1
- 5x+6<6 or 9x−2>−11
- 2(3x−1)<−16 or 3(1−2x)<−15
- 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?
- 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.
- 5−3(12−|2−52|)
- (−12)2−(3−2|−34|)3
- −7√60
- 53√−32
- Find the diagonal of a square with sides measuring 6 centimeters.
- Answer
-
1. 38
3. −14√15
5. 6√2 centimeters
Exercise 1.E.20
Simplify
- −5x2yz−1(3x3y−2z)
- (−2a−4b2ca−3b0c2)−3
- 2(3a2b2+2ab−1)−a2b2+2ab−1
- (x2−6x+9)−(3x2−7x+2)
- (2x−3)3
- (3a−b)(9a2+3ab+b2)
- 6x4−17x3+16x2−18x+132x−3
- Answer
-
2. −a3c38b6
4. −2x2+x+7
6. 27a3−b3
Exercise 1.E.21
Solve.
- 45x−215=2
- 34(8x−12)−12(2x−10)=16
- 12−5(3x−1)=2(4x+3)
- 12(12x−2)+5=4(32x−8)
- Solve for y:ax+by=c
- Answer
-
1. 83
3. 1123
5. y=c−axb
Exercise 1.E.22
Solve. Graph the solutions on a number line and give the corresponding interval notation.
- 2(3x−5)−(7x−3)≥0
- 2(4x−1)−4(5+2x)<−10
- −6≤14(2x−8)<4
- 3x−7>14 or 3x−7<−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.
- Degrees Fahrenheit F is given by the formula F=95C+32 where C represents degrees Celsius. What is the Fahrenheit equivalent to 35° Celsius?
- 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.
- 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?
- 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.