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2.8E: Exercises for Section 2.8

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In Exercises 1 - 8, use both interval and set-builder notation to describe the intersection of the two intervals shown on the graph. Also, sketch the graph of the intersection on the real number line.

Exercise 1

Screen Shot 2019-07-29 at 10.25.01 PM.png

Answer

The intersection is the set of points that are in both intervals (shaded on both graphs). Graph of the intersection:

Screen Shot 2019-08-05 at 10.52.32 AM.png

Set-Builder Notation: {x|x1}

Interval Notation: [1,)

Exercise 2

Screen Shot 2019-07-29 at 10.26.55 PM.png

Exercise 3

Screen Shot 2019-07-29 at 10.27.31 PM.png

Answer

There are no points that are in both intervals (shaded in both), so there is no intersection. Graph of the intersection:

Screen Shot 2019-08-05 at 10.53.41 AM.png

no intersection

Set-Builder Notation:  {}  

Interval Notation: none

Exercise 4

Screen Shot 2019-07-29 at 10.28.18 PM.png

Exercise 5

Screen Shot 2019-07-29 at 10.29.22 PM.png

Answer

The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:

Screen Shot 2019-08-05 at 10.54.24 AM.png

Set-Builder Notation: {x|6x2}

Interval Notation: [6,2]

Exercise 6

Screen Shot 2019-07-29 at 10.30.25 PM.png

Exercise 7

Screen Shot 2019-07-29 at 10.31.23 PM.png

Answer

The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:

Screen Shot 2019-08-05 at 10.55.19 AM.png

Set-Builder Notation: {x|x9}

Interval Notation: [9,)

Exercise 8

Screen Shot 2019-07-29 at 10.33.57 PM.png

 

In Exercises 9 - 16, use both interval notation and set-builder notation to describe the union of the two intervals shown on the graph. Also, sketch the graph of the union on the real number line.

Exercise 9

Screen Shot 2019-07-29 at 10.37.45 PM.png

Answer

The union is the set of all points that are in one interval or the other (shaded in either graph). Graph of the union:

Screen Shot 2019-08-05 at 10.58.47 AM.png

Set-Builder Notation: {x|x8}

Interval Notation: (,8]

Exercise 10

Screen Shot 2019-07-29 at 10.39.08 PM.png

Exercise 11

Screen Shot 2019-07-29 at 10.39.38 PM.png

Answer

The union is the set of all points that are in one interval or the other (shaded in either graph). Graph of the union:

Screen Shot 2019-08-05 at 11.01.44 AM.png

Set-Builder Notation: {x|x9 or x>15}

Interval Notation: (,9](15,)

Exercise 12

Screen Shot 2019-07-29 at 10.40.55 PM.png

Exercise 13

Screen Shot 2019-07-29 at 10.42.41 PM.png

Answer

The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:

Screen Shot 2019-08-05 at 11.03.02 AM.png

Set-Builder Notation: {x|x<3}

Interval Notation: (,3)

Exercise 14

Screen Shot 2019-07-29 at 10.44.52 PM.png

Exercise 15

Screen Shot 2019-07-29 at 10.45.57 PM.png

Answer

The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:

Screen Shot 2019-08-05 at 11.04.38 AM.png

Set-Builder Notation: {x|x9}

Interval Notation: [9,)

Exercise 16

Screen Shot 2019-07-29 at 10.46.54 PM.png

 

In Exercises 17 - 32, use interval notation to describe the given set. Also, sketch the graph of the set on the real number line.

Exercise 17

{x|x6 and x>5}

Answer

This set is the same as {x|x>5}, which is (5,) in interval notation. Graph of the set:

Screen Shot 2019-08-05 at 11.06.12 AM.png

Exercise 18

{x|x6 and x4}

Exercise 19

{x|x1 or x<3}

Answer

Every real number is in one or the other of the two intervals. Therefore, the set is the set of all real numbers (,). Graph of the set:

Screen Shot 2019-08-05 at 11.07.04 AM.png

Exercise 20

{x|x>7 and x>4}

Exercise 21

{x|x1 or x>6}

Answer

This set is the same as {x|x1}, which is [1,) in interval notation. Graph of the set:

Screen Shot 2019-08-05 at 11.10.10 AM.png

Exercise 22

{x|x7 or x<2}

Exercise 23

{x|x6 or x>3}

Answer

This set is the same as {x|x>3}, which is (3,) in interval notation. Graph of the set:

Screen Shot 2019-08-05 at 11.12.01 AM.png

Exercise 24

{x|x1 or x>0}

Exercise 25

{x|x<2 and x<7}

Answer

This set is the same as {x|x<7}, which is (,7) in interval notation. Graph of the set:

Screen Shot 2019-08-05 at 11.13.07 AM.png

Exercise 26

{x|x3 and x<5}

Exercise 27

{x|x3 or x4}

Answer

This set is the union of two intervals, (,3][4,). Graph of the set:

Screen Shot 2019-08-05 at 11.14.04 AM.png

Exercise 28

{x|x<11 or x8}

Exercise 29

{x|x5 and x1}

Answer

There are no numbers that satisfy both inequalities. Thus, there is no intersection. Graph of the set:

Screen Shot 2019-08-05 at 11.14.56 AM.png

Exercise 30

{x|x<5 or x<10}

Exercise 31

{x|x5 and x1}

Answer

This set is the same as {x|1x5}, which is [−1, 5] in interval notation. Graph of the set

Screen Shot 2019-08-05 at 11.17.54 AM.png

Exercise 32

{x|x>3 and x<6}

 

In Exercises 33 - 44, solve the inequality. Express your answer in both interval and set-builder notations, and graph the solution on a number line. Answers include worked out solutions.

Exercise 33

8x316x1

Answer

8x316x18x+16x1+38x2x14

Thus, the solution interval is

Screen Shot 2019-08-07 at 10.21.02 PM.png

Set-Builder Notation: {x|x14}

Interval Notation: (,14]

Exercise 34

6x6>3x+3

Exercise 35

12x+53x4

Answer

12x+53x412x+3x459x9x1

Thus, the solution interval is

Screen Shot 2019-08-08 at 10.56.57 PM.png

Set-Builder Notation: \(\\{x \, | \, x\geq 1\}\)

Interval Notation: [1,)

Exercise 36

7x+32x8

Exercise 37

11x9<3x+1

Answer

11x9<3x+111x+3x<1+98x<10x>54

Thus, the solution interval is

Screen Shot 2019-08-08 at 11.00.01 PM.png

Set-Builder Notation: {x|x>54}

Interval Notation: (54,)

Exercise 38

4x84x5

Exercise 39

4x5>5x7

Answer

4x5>5x74x5x>7+5x>2x<2
Thus, the solution interval is

Screen Shot 2019-08-08 at 11.02.23 PM.png

Set-Builder Notation: {x|x<2}

Interval Notation: (,2)

Exercise 40

14x+4>6x+8

Exercise 41

2x1>7x+2

Answer

2x1>7x+22x7x>2+15x>3x<35
Thus, the solution interval is

Screen Shot 2019-08-08 at 11.04.53 PM.png

Set-Builder Notation: {x|x<35}

Interval Notation: (,35)

Exercise 42

3x2>4x9

Exercise 43

3x+3<11x3

Answer

3x+3<11x33x+11x<338x<6x<34
Thus, the solution interval is

Screen Shot 2019-08-08 at 11.07.23 PM.png

Set-Builder Notation: {x|x<34}

Interval Notation: (,34)

Exercise 44

6x+3<8x+8

 

In Exercises 45-53, solve the compound inequality. Express your answer in both interval and set-builder notations, and graph the solution on a number line.  Worked out solutions are provided in the Answers for #45 and #47. Only the interval notation version of the answer is given for Answers in #49-82.

Exercise 45

2x1<4 or 7x+14

Answer

2x1<4 or 7x+142x<5or7x5x<52orx57

Screen Shot 2019-08-08 at 11.11.56 PM.png

For the union, shade anything shaded in either graph. The solution is the set of all real numbers. (,).

Screen Shot 2019-08-08 at 11.13.59 PM.png

Set-Builder Notation: {x|xR}

Interval Notation: (,)

Exercise 46

8x+9<3 and 7x+1>3

Exercise 47

6x4<4 and 3x+75

Answer

6x4<4 and 3x+756x<0and3x12x>0andx40<x4

Screen Shot 2019-08-08 at 11.21.51 PM.png

The intersection is all points shaded in both graphs, so the solution is

Screen Shot 2019-08-08 at 11.23.30 PM.png

Set-Builder Notation: {x|0<x4}

Interval Notation: (0,4]

Exercise 48

3x+38 and 3x6>6

Exercise 49

8x+51 and 4x2>1

Answer

No Solution
Set-Builder Notation: {}

Exercise 50

x1<7 and 6x98

Exercise 51

3x+85 or 2x43

Answer

No Solution
Set-Builder Notation: {}

Exercise 52

6x7<3 and 8x3

Exercise 53

9x99 and 5x>1

Answer

Interval Notation: (15,2]

Exercise 2.8E.54

7x+3<3 or 8x2

Exercise 2.8E.55

3x5<4 and x+9>3

Answer

Interval Notation: (,3)

Exercise 2.8E.56

8x6<5 or 4x13

Exercise 2.8E.57

9x+35 or 2x49

Answer

Interval Notation: (,89)

Exercise 2.8E.58

7x+6<4 or 7x5>7

Exercise 2.8E.59

4x22 or 3x93

Answer

Add texts here. (,1][4,)

Exercise 2.8E.60

5x+5<4 or 5x55

Exercise 2.8E.61

5x+1<6 and 3x+9>4

Answer

Interval Notation: (133,75)

Exercise 2.8E.62

7x+2<5 or 6x97

Exercise 2.8E.63

7x7<2 and 3x3

Answer

Interval Notation: [1,)

Exercise 2.8E.64

4x+1<0 or 8x+6>9

Exercise 2.8E.65

7x+8<3 and 8x+39

Answer

No Solution
Set-Builder Notation: {}

Exercise 2.8E.66

3x<2 and 7x83

Exercise 2.8E.67

5x+22 and 6x+23

Answer

No Solution
Set-Builder Notation: {}

Exercise 2.8E.68

4x18 or 3x9>0

Exercise 2.8E.69

2x51 and 4x+7>7

Answer

(0,3]

Exercise 2.8E.70

3x+1<0 or 5x+5>8

Exercise 2.8E.71

8x+79 or 5x+6>2

Answer

(,)

Exercise 2.8E.72

x65 and 6x2>3

Exercise 2.8E.73

4x8<4 or 4x+2>3

Answer

(,)

Exercise 2.8E.74

9x5<2 or 8x56

Exercise 2.8E.75

9x53 or x+1>3

Answer

(29,)

Exercise 2.8E.76

5x36 and 2x16

Exercise 2.8E.77

17x32

Answer

Interval Notation: [57,27]

Exercise 2.8E.78

0<5x5<9

Exercise 79

5<9x36

Answer

Graph of the solution:     MTH098 Section 2.8E #79.png

Set-Builder Notation: {x|89<x1}

Interval Notation: (89,1]

Exercise 2.8E.80

6<7x+32

Exercise 2.8E.81

2<7x+6<6

Answer

Interval Notation: (0,87)

Exercise 82

9<2x+51

In Exercises 83-94, solve the given inequality for x. Graph the solution set on a number line, then use interval and setbuilder notation to describe the solution set. Only the interval notation version of the answer is given for Answers in #83-94.

Exercise 2.8E.83

13<x2+14<13

Answer

Interval Notation: (76,16)

Exercise 2.8E.84

15<x214<15

Exercise 2.8E.85

12<13x2<12

Answer

Interval Notation: (13,53)

Exercise 86

2312x523

Exercise 2.8E.87

1<xx+15<2

Answer

Interval Notation: (1,114)

Exercise 2.8E.88

2<x2x13<4

Exercise 2.8E.89

2<x+12x+132

Answer

Interval Notation: (13,11]

Exercise 2.8E.90

3<x132x152

Exercise 2.8E.91

x<4x<5

Answer

Interval Notation: (1,2)

Exercise 2.8E.92

x<2x+37

Exercise 2.8E.93

x<x+511

Answer

Interval Notation: (52,6]

Exercise 2.8E.94

2x<3x8

 

More Practice with Compound Inequalities

Exercise 2.8E.95

For each graph below, describe the interval (a) using set-builder notation and (b) using interval notation. 

  1.
WeChatb5d2ec2c0d59c1eea3469a07a6e3eeb3.png

  2.
    Screen Shot 2019-08-05 at 10.49.22 AM.png

  3.
   Screen Shot 2019-08-05 at 10.50.50 AM.png

   4.
  Screen Shot 2019-08-05 at 10.54.24 AM.png

5.
 Screen Shot 2019-08-05 at 11.17.54 AM.png

6.
 Screen Shot 2019-08-08 at 11.23.30 PM.png

7.
 WeChata8b836e07fe50d89bd7fc775102533f2.pngx

8.
 WeChat4bb84b27707520b9ef125fb6f5e8b38a.png

Answer
1.  (a) {x|3x4}         (b) [3,4]
3.  (a) {x|4<x<1}      (b) (4,1)
5.  (a) {x|1x5}      (b) [1,5]
7.  (a) {x|1<x5}      (b) (1,5]

 

 

 


2.8E: Exercises for Section 2.8 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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