Algebra 1B Section 5-4

Section 5-4
Compound Inequalities

Monday, October 14, 13

Essential Question

✤

How do you solve compound inequalities?


Vocabulary
1. Compound Inequality:

2. Intersection:

3. Union:


Vocabulary
1. Compound Inequality: Two inequalities that are utilized at the same
time

2. Intersection:

3. Union:


Vocabulary
time

2. Intersection: A compound inequality the connects the two
inequalities using and; Only true if BOTH inequalities are true

3. Union:


Vocabulary
time

2. Intersection: A compound inequality the connects the two
inequalities using and; Only true if BOTH inequalities are true

3. Union: A compound inequality the connects the two inequalities
using or; True if EITHER inequality is true


Example 1
Solve and graph the solution set.

a. 7 < x + 2 ≤ 11


Example 1

a. 7 < x + 2 ≤ 11
7< x+2


Example 1

a. 7 < x + 2 ≤ 11
7< x+2


x + 2 ≤ 11

Example 1

a. 7 < x + 2 ≤ 11
7< x+2


and

x + 2 ≤ 11

Example 1

a. 7 < x + 2 ≤ 11
7< x+2
−2 −2


and

x + 2 ≤ 11

Example 1

a. 7 < x + 2 ≤ 11
7< x+2
−2 −2
5<x


and

x + 2 ≤ 11

Example 1

a. 7 < x + 2 ≤ 11
7< x+2
−2 −2
5<x


and

x + 2 ≤ 11
−2 −2

Example 1

a. 7 < x + 2 ≤ 11
7< x+2
−2 −2
5<x


and

x + 2 ≤ 11
−2 −2
x≤9

Example 1

a. 7 < x + 2 ≤ 11
7< x+2
−2 −2
5<x

0

and

5

x + 2 ≤ 11
−2 −2
x≤9

10

Example 1

a. 7 < x + 2 ≤ 11
7< x+2
−2 −2
5<x

0

and

{x|5 < x ≤ 9}

5

x + 2 ≤ 11
−2 −2
x≤9

10

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30


Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25


Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25


12 − 9k ≥ 30

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25


or

12 − 9k ≥ 30

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7


or

12 − 9k ≥ 30

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32


or

12 − 9k ≥ 30

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4


or

12 − 9k ≥ 30

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8


or

12 − 9k ≥ 30

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8


or

12 − 9k ≥ 30
−12
−12

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8


or

12 − 9k ≥ 30
−12
−12
−9k ≥ 18

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8


or

12 − 9k ≥ 30
−12
−12
−9k ≥ 18
−9 −9

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8


or

12 − 9k ≥ 30
−12
−12
−9k ≥ 18
−9 −9
k ≤ −2

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8

-2

0

12 − 9k ≥ 30
−12
−12
−9k ≥ 18
−9 −9
k ≤ −2

or

2

4

6

8

Example 1

b. 4k − 7 ≤ 25 or 12 − 9k ≥ 30
4k − 7 ≤ 25
+7 +7
4k ≤ 32
4 4
k≤8

-2

0

12 − 9k ≥ 30
−12
−12
−9k ≥ 18
−9 −9
{k|k ≤ 8}
k ≤ −2
or

2

4

6

8

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5


Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9


Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9


3y + 4 < −5

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9


or

3y + 4 < −5

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5


or

3y + 4 < −5

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4


or

3y + 4 < −5

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1


or

3y + 4 < −5

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4


or

3y + 4 < −5

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4


or

3y + 4 < −5
−4 −4

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4


or

3y + 4 < −5
−4 −4
3y < −9

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4


or

3y + 4 < −5
−4 −4
3y < −9
3 3

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4


or

3y + 4 < −5
−4 −4
3y < −9
3 3
y < −3

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4

-6

3y + 4 < −5
−4 −4
3y < −9
3 3
y < −3

or

-4

0

2

4

6

Example 1

c. − y + 5 ≥ 9 or 3y + 4 < −5
3y + 4 < −5
−4 −4
3y < −9
3 3
{y|y < −3}
y < −3

−y + 5 ≥ 9
−5 −5
−y ≥ 4
−1 −1
y ≤ −4

-6

or

-4

0

2

4

6

Example 1

d. d − 3 < 6d + 12 < 2d + 32


Example 1

d. d − 3 < 6d + 12 < 2d + 32
d − 3 < 6d + 12


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
d − 3 < 6d + 12


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−3 < 5d + 12


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−3 < 5d + 12
−12
−12


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−3 < 5d + 12
−12
−12
−15 < 5d


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−3 < 5d + 12
−12
−12
−15 < 5d
5 5


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−3 < 5d + 12
−12
−12
−15 < 5d
5 5
−3 < d


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−3 < 5d + 12
−12
−12
−15 < 5d
5 5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
−12
−12
−15 < 5d
5 5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−15 < 5d
5 5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−12 −12
−15 < 5d
5 5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−12 −12
−15 < 5d
4d < 20
5 5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−12 −12
−15 < 5d
4d < 20
4
4
5 5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−12 −12
−15 < 5d
4d < 20
4
4
5 5
d<5
−3 < d d > −3


Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−12 −12
−15 < 5d
4d < 20
4
4
5 5
d<5
−3 < d d > −3
-6

-4

0

2

4

6

Example 1

d. d − 3 < 6d + 12 < 2d + 32
6d + 12 < 2d + 32
and
d − 3 < 6d + 12
−d
−d
−2d
−2d
−3 < 5d + 12
4d + 12 < 32
−12
−12
−12 −12
−15 < 5d
4d < 20
{d|−3 < d < 5}
4
4
5 5
d<5
−3 < d d > −3
-6

-4

0

2

4

6

Example 2
A ski resort has several types of hotel rooms and cabins. The hotel
rooms cost at most $89 per night and the cabins cost at least $109 per
night. Write and graph a compound inequality that describes the
amount a guest would pay per night at the resort.


Example 2
c = cost per night


Example 2
c = cost per night
Room


Example 2
c = cost per night
Room


Cabin

Example 2
c = cost per night
Room

c ≤ 89


Cabin

Example 2
c = cost per night
Room

c ≤ 89


Cabin

c ≥ 109

Example 2
c = cost per night
Room

Cabin

c ≤ 89

c ≥ 109

86

90

94

98

102

106

110

Example 2
c = cost per night
Room

Cabin

c ≤ 89

c ≥ 109

{c|c ≤ 89 or c ≥ 109}
86

90

94

98

102

106

110

Example 3
Write a compound inequality and solve.
a. Eight less than a number is no more than fourteen and no less than
ﬁve.


Example 3
ﬁve.
n = number


Example 3
ﬁve.
n = number

n − 8 ≤ 14


Example 3
ﬁve.
n = number

n − 8 ≤ 14


n−8≥ 5

Example 3
ﬁve.
n = number

n − 8 ≤ 14
+8 +8


n−8≥ 5

Example 3
ﬁve.
n = number

n − 8 ≤ 14
+8 +8
n ≤ 22


n−8≥ 5

Example 3
ﬁve.
n = number

n − 8 ≤ 14
+8 +8
n ≤ 22


n−8≥ 5
+8 +8

Example 3
ﬁve.
n = number

n − 8 ≤ 14
+8 +8
n ≤ 22


n−8≥ 5
+8 +8
n ≥ 13

Example 3
ﬁve.
n = number

n − 8 ≤ 14
+8 +8
n ≤ 22

n−8≥ 5
+8 +8
n ≥ 13
{n|13 ≤ n ≤ 22}


Example 3
b. The product of negative ﬁve and a number is greater than thirty ﬁve
or less than ten.


Example 3
or less than ten.
n = number


Example 3
or less than ten.
n = number

−5n > 35


Example 3
or less than ten.
n = number

−5n > 35


−5n < 10

Example 3
or less than ten.
n = number

−5n > 35
−5 −5


−5n < 10

Example 3
or less than ten.
n = number

−5n > 35
−5 −5
n < −7


−5n < 10

Example 3
or less than ten.
n = number

−5n > 35
−5 −5
n < −7


−5n < 10
−5 −5

Example 3
or less than ten.
n = number

−5n > 35
−5 −5
n < −7


−5n < 10
−5 −5
n > −2

Example 3
or less than ten.
n = number

−5n > 35
−5 −5
n < −7

−5n < 10
−5 −5
n > −2

{n|n < −7 or n > −2}

Summarizer
Write a compound inequality for which the graph is the empty set and
one for which the graph is the set of all real numbers.


Problem Sets


Problem Sets
Problem Set 1: p. 306 #1-5 all, 7-15 odd
Problem Set 2: p. 306 #16, 17-27 odd, 28, 29, 31, 33-36 all

“It is always easier to believe than to deny. Our minds are naturally aﬃrmative."
- John Burroughs

Algebra 1B Section 5-4

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Algebra 1B Section 5-4