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Solving Linear Equations and Inequalities: Exercise Supplement

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))). This module contains the exercise supplement for the chapter "Solving Linear Equations and Inequalities".

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Exercise Supplement

Solving Equations ((Reference)) - Further Techniques in Equation Solving ((Reference))

Solve the equations for the following problems.

Exercise 1

y+3=11 y+3=11

Solution

y=8 y=8

Exercise 2

a−7=4 a−7=4

Exercise 3

r−1=16 r−1=16

Solution

r=17 r=17

Exercise 4

a+2=0 a+2=0

Exercise 5

x+6=−4 x+6=−4

Solution

x=−10 x=−10

Exercise 6

x−5=−6 x−5=−6

Exercise 7

x+8=8 x+8=8

Solution

x=0 x=0

Exercise 8

y−4=4 y−4=4

Exercise 9

2x=32 2x=32

Solution

x=16 x=16

Exercise 10

4x=24 4x=24

Exercise 11

3r=−12 3r=−12

Solution

r=−4 r=−4

Exercise 12

6m=−30 6m=−30

Exercise 13

−5x=−30 −5x=−30

Solution

x=6 x=6

Exercise 14

−8y=−72 −8y=−72

Exercise 15

−x=6 −x=6

Solution

x=−6 x=−6

Exercise 16

−y=−10 −y=−10

Exercise 17

3x+7=19 3x+7=19

Solution

x=4 x=4

Exercise 18

6x−1=29 6x−1=29

Exercise 19

4x+2=−2 4x+2=−2

Solution

x=−1 x=−1

Exercise 20

6x−5=−29 6x−5=−29

Exercise 21

8x+6=−10 8x+6=−10

Solution

x=−2 x=−2

Exercise 22

9a+5=−22 9a+5=−22

Exercise 23

m 6 +4=8 m 6 +4=8

Solution

m=24 m=24

Exercise 24

b 5 −2=5 b 5 −2=5

Exercise 25

y 9 =54 y 9 =54

Solution

y=486 y=486

Exercise 26

a −3 =−17 a −3 =−17

Exercise 27

c 6 =15 c 6 =15

Solution

c=90 c=90

Exercise 28

3a 4 =9 3a 4 =9

Exercise 29

4y 5 =−12 4y 5 =−12

Solution

y=−15 y=−15

Exercise 30

r 4 =7 r 4 =7

Exercise 31

6a −5 =11 6a −5 =11

Solution

a=− 55 6 a=− 55 6

Exercise 32

9x 7 =6 9x 7 =6

Exercise 33

c 2 −8=0 c 2 −8=0

Solution

c=16 c=16

Exercise 34

m −5 +4=−1 m −5 +4=−1

Exercise 35

x 7 −15=−11 x 7 −15=−11

Solution

x=28 x=28

Exercise 36

3x 4 +2=14 3x 4 +2=14

Exercise 37

3r+2 5 =−1 3r+2 5 =−1

Solution

r=− 7 3 r=− 7 3

Exercise 38

6x−1 7 =−3 6x−1 7 =−3

Exercise 39

4x−3 6 +2=−6 4x−3 6 +2=−6

Solution

x=− 45 4 x=− 45 4

Exercise 40

y−21 8 =−3 y−21 8 =−3

Exercise 41

4(x+2)=20 4(x+2)=20

Solution

x=3 x=3

Exercise 42

−2(a−3)=16 −2(a−3)=16

Exercise 43

−7(2a−1)=63 −7(2a−1)=63

Solution

a=−4 a=−4

Exercise 44

3x+7=5x−21 3x+7=5x−21

Exercise 45

−(8r+1)=33 −(8r+1)=33

Solution

r=− 17 4 r=− 17 4

Exercise 46

Solve I=prt for t. Find the value of t when I=3500, P=3000, and r=0.05. Solve I=prt for t. Find the value of t when I=3500, P=3000, and r=0.05.

Exercise 47

Solve A=LW for W. Find the value of W when A=26 and L=2. Solve A=LW for W. Find the value of W when A=26 and L=2.

Solution

W=13 W=13

Exercise 48

Solve p=mv for m. Find the value of m when p=4240 and v=260. Solve p=mv for m. Find the value of m when p=4240 and v=260.

Exercise 49

Solve P=R−C for R. Find the value of R when P=480 and C=210. Solve P=R−C for R. Find the value of R when P=480 and C=210.

Solution

R=690 R=690

Exercise 50

Solve P= nRT V  for n. Solve P= nRT V  for n.

Exercise 51

Solve y=5x+8 for x. Solve y=5x+8 for x.

Solution

x= y−8 5 x= y−8 5

Exercise 52

Solve 3y−6x=12 for y. Solve 3y−6x=12 for y.

Exercise 53

Solve 4y+2x+8=0 for y. Solve 4y+2x+8=0 for y.

Solution

y=− 1 2 x−2 y=− 1 2 x−2

Exercise 54

Solve k= 4m+6 7  for m. Solve k= 4m+6 7  for m.

Exercise 55

Solve t= 10a−3b 2c  for b. Solve t= 10a−3b 2c  for b.

Solution

b=− 2ct−10a 3 b=− 2ct−10a 3

Application I - Translating from Verbal to Mathetical Expressions ((Reference))

For the following problems, translate the phrases or sentences to mathematical expressions or equations.

Exercise 56

A quantity less eight.

Exercise 57

A number, times four plus seven.

Solution

x( 4+7 ) x( 4+7 )

Exercise 58

Negative ten minus some number.

Exercise 59

Two fifths of a number minus five.

Solution

2 5 x−5 2 5 x−5

Exercise 60

One seventh of a number plus two ninths of the number.

Exercise 61

Three times a number is forty.

Solution

3x=40 3x=40

Exercise 62

Twice a quantity plus nine is equal to the quantity plus sixty.

Exercise 63

Four times a number minus five is divided by seven. The result is ten more than the number.

Solution

( 4x−5 ) 7 =x+10 ( 4x−5 ) 7 =x+10

Exercise 64

A number is added to itself five times, and that result is multiplied by eight. The entire result is twelve.

Exercise 65

A number multiplied by eleven more than itself is six.

Solution

x( x+11 )=6 x( x+11 )=6

Exercise 66

A quantity less three is divided by two more than the quantity itself. The result is one less than the original quantity.

Exercise 67

A number is divided by twice the number, and eight times the number is added to that result. The result is negative one.

Solution

x 2x +8x=−1 x 2x +8x=−1

Exercise 68

An unknown quantity is decreased by six. This result is then divided by twenty. Ten is subtracted from this result and negative two is obtained.

Exercise 69

One less than some number is divided by five times the number. The result is the cube of the number.

Solution

x−1 5x = x 3 x−1 5x = x 3

Exercise 70

Nine less than some number is multiplied by the number less nine. The result is the square of six times the number.

Application II - Solving Problems ((Reference))

For the following problems, find the solution.

Exercise 71

This year an item costs $106, an increase of $10 over last year’s price. What was last year’s price?

Solution

last year's price=$96 last year's price=$96

Exercise 72

The perimeter of a square is 44 inches. Find the length of a side.

Exercise 73

Nine percent of a number is 77.4. What is the number?

Solution

x=860 x=860

Exercise 74

Two consecutive integers sum to 63. What are they?

Exercise 75

Four consecutive odd integers add to 56. What are they?

Solution

x=11 x+2=13 x+4=15 x+6=17 x=11 x+2=13 x+4=15 x+6=17

Exercise 76

If twenty-one is subtracted from some number and that result is multiplied by two, the result is thirty-eight. What is the number?

Exercise 77

If 37% more of a quantity is 159.1, what is the quantit?

Solution

x=116.13139 x=116.13139

Exercise 78

A statistician is collecting data to help her estimate the number of pickpockets in a certain city. She needs 108 pieces of data and is 3 4 3 4 done. How many pieces of data has she collected?

Exercise 79

The statistician in problem 78 is eight pieces of data short of being 5 6 5 6 done. How many pieces of data has she collected?

Solution

82 pieces of data 82 pieces of data

Exercise 80

A television commercial advertises that a certain type of light bulb will last, on the average, 200 hours longer than three times the life of another type of bulb. If consumer tests show that the advertised bulb lasts 4700 hours, how many hours must the other type of bulb last for the advertiser’s claim to be valid?

Linear inequalities in One Variable ((Reference))

Solve the inequalities for the following problems.

Exercise 81

y+3<15 y+3<15

Solution

y<12 y<12

Exercise 82

x−6≥12 x−6≥12

Exercise 83

4x+3>23 4x+3>23

Solution

x>5 x>5

Exercise 84

5x−14<1 5x−14<1

Exercise 85

6a−6≤−27 6a−6≤−27

Solution

a≤− 7 2 a≤− 7 2

Exercise 86

−2y≥14 −2y≥14

Exercise 87

−8a≤−88 −8a≤−88

Solution

a≥11 a≥11

Exercise 88

x 7 >−2 x 7 >−2

Exercise 89

b −3 ≤4 b −3 ≤4

Solution

b≥−12 b≥−12

Exercise 90

2a 7 <6 2a 7 <6

Exercise 91

16c 3 ≥−48 16c 3 ≥−48

Solution

c≥−9 c≥−9

Exercise 92

−4c+3≤5 −4c+3≤5

Exercise 93

−11y+4>15 −11y+4>15

Solution

y<−1 y<−1

Exercise 94

3(4x−5)>−6 3(4x−5)>−6

Exercise 95

−7(8x+10)+2<−32 −7(8x+10)+2<−32

Solution

x>− 9 14 x>− 9 14

Exercise 96

5x+4≥7x+16 5x+4≥7x+16

Exercise 97

−x−5<3x−11 −x−5<3x−11

Solution

x> 3 2 x> 3 2

Exercise 98

4(6x+1)+2≥−3(x−1)+4 4(6x+1)+2≥−3(x−1)+4

Exercise 99

−(5x+6)+2x−1<3(1−4x)+11 −(5x+6)+2x−1<3(1−4x)+11

Solution

x< 7 3 x< 7 3

Exercise 100

What numbers satisfy the condition: nine less than negative four times a number is strictly greater than negative one?

Linear Equations in Two Variables ((Reference))

Solve the equations for the following problems.

Exercise 101

y=−5x+4, if x=−3 y=−5x+4, if x=−3

Solution

( −3,19 ) ( −3,19 )

Exercise 102

y=−10x+11, if x=−1 y=−10x+11, if x=−1

Exercise 103

3a+2b=14, if b=4 3a+2b=14, if b=4

Solution

( 2,4 ) ( 2,4 )

Exercise 104

4m+2k=30, if m=8 4m+2k=30, if m=8

Exercise 105

−4r+5s=−16, if s=0 −4r+5s=−16, if s=0

Solution

( 4,0 ) ( 4,0 )

Exercise 106

y=−2(7x−4), if x=−1 y=−2(7x−4), if x=−1

Exercise 107

−4a+19=2(b+6)−5, if b=−1 −4a+19=2(b+6)−5, if b=−1

Solution

( 7 2 ,−1 ) ( 7 2 ,−1 )

Exercise 108

6(t+8)=−(a−5), if a=10 6(t+8)=−(a−5), if a=10

Exercise 109

−(a+b)=5, if a=−5 −(a+b)=5, if a=−5

Solution

( −5,0 ) ( −5,0 )

Exercise 110

−a(a+1)=2b+1, if a=−2 −a(a+1)=2b+1, if a=−2

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