Interval Notation and Absolute Value
Transcript of Interval Notation and Absolute Value
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Intervals are sets of real numbers. The notation uses square and
round brackets to show these sets of numbers.
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Intervals are sets of real numbers. The notation uses square and
round brackets to show these sets of numbers.
Round bracket – go up to but do not include this number in the set
![Page 4: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/4.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Intervals are sets of real numbers. The notation uses square and
round brackets to show these sets of numbers.
( 3 , 7 ) - this interval would include all numbers between 3
and 7, but NOT 3 or 7.
Round bracket – go up to but do not include this number in the set
![Page 5: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/5.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Intervals are sets of real numbers. The notation uses square and
round brackets to show these sets of numbers.
Square bracket – include this number in the set
( 3 , 7 ) - this interval would include all numbers between 3
and 7, but NOT 3 or 7.
Round bracket – go up to but do not include this number in the set
![Page 6: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/6.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Intervals are sets of real numbers. The notation uses square and
round brackets to show these sets of numbers.
Square bracket – include this number in the set
( 3 , 7 ) - this interval would include all numbers between 3
and 7, but NOT 3 or 7.
Round bracket – go up to but do not include this number in the set
[ 3 , 7 ] - this interval would include all numbers from 3 to 7..
![Page 7: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/7.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
When working with equations containing an inequality, the symbols
for the inequality determine how you graph and represent the
solution as an interval.
Round bracket - less than ( < ) , greater than ( > )
![Page 8: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/8.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
- open circle on a graph
When working with equations containing an inequality, the symbols
for the inequality determine how you graph and represent the
solution as an interval.
![Page 9: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/9.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket – less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
When working with equations containing an inequality, the symbols
for the inequality determine how you graph and represent the
solution as an interval.
![Page 10: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/10.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
When working with equations containing an inequality, the symbols
for the inequality determine how you graph and represent the
solution as an interval.
- closed circle on a graph
![Page 11: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/11.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE : Solve and graph and show your answer as an interval 753 x
![Page 12: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/12.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE : Solve and graph and show your answer as an interval 753 x
4
123
55
753
x
x
x
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE : Solve and graph and show your answer as an interval 753 x
4
123
55
753
x
x
x
4 graph
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE : Solve and graph and show your answer as an interval 753 x
4
123
55
753
x
x
x
4
) ,4 (
graph
interval
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
5123 x
![Page 16: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/16.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
5123 x
31
622
11 1
5123
x
x
x
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
5123 x
31
622
11 1
5123
x
x
x
This results in two graphs…
x < 3
x ≥ -1
3 - 1
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
5123 x
31
622
11 1
5123
x
x
x
The solution set is
where the two graphs
overlap ( share )
3 - 1
![Page 19: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/19.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
5123 x
31
622
11 1
5123
x
x
x
The solution set is
where the two graphs
overlap ( share )
3 - 1
[ -1 , 3 ) interval
![Page 20: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/20.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 3 : Solve and graph and show your answer
as an interval
01272 xx
![Page 21: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/21.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
3,4
034
01272
x
xx
xx
01272 xx
![Page 22: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/22.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
3,4
034
01272
x
xx
xx
These are our critical points
01272 xx
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
3,4
034
01272
x
xx
xx
These are our critical points
- 3 - 4
01272 xx
Graph the critical points and then
use a test point to find “true/false”
![Page 24: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/24.jpg)
CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
3,4
034
01272
x
xx
xx
These are our critical points
- 3 - 4
01272 xx
Graph the critical points and then
use a test point to find “true/false”
TEST x = 0
TRUE 012
01200
0120702
0
TRUE FALSE TRUE
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CALCULUS 1 – Algebra review
Intervals and Interval Notation
Round bracket - less than ( < ) , greater than ( > )
Square bracket - less than or equal to ( ≤ ), greater than or equal
to ( ≥ )
- open circle on a graph
- closed circle on a graph
EXAMPLE # 2 : Solve and graph and show your answer
as an interval
3,4
034
01272
x
xx
xx
These are our critical points
- 3 - 4
01272 xx
Graph the critical points and then
use a test point to find “true/false”
TEST x = 0
TRUE 012
01200
0120702
0
TRUE FALSE TRUE
,34,interval
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CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
![Page 27: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/27.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 1 : Solve 752 x
![Page 28: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/28.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 1 : Solve 752 x
16
2
2
2
2
2
12
2212
55 5
7527
752
x
x
x
x
x
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CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 2 : Solve 6222
xx
![Page 30: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/30.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 2 : Solve 6222
xx
Remember u substitution
from pre-calc ?
023
06
6
2Let
2
2
uu
uu
uu
xu
![Page 31: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/31.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 2 : Solve 6222
xx
Remember u substitution
from pre-calc ?
023
06
6
2Let
2
2
uu
uu
uu
xu
22 and 32
022 and 032
xx
xx
![Page 32: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/32.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 2 : Solve 6222
xx
Remember u substitution
from pre-calc ?
023
06
6
2Let
2
2
uu
uu
uu
xu
22 and 32
022 and 032
xx
xx
Can’t have an absolute value equal
to a negative answer
![Page 33: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/33.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 2 : Solve 6222
xx
Remember u substitution
from pre-calc ?
023
06
6
2Let
2
2
uu
uu
uu
xu
51
323
32
032
x
x
x
x
Now solve the
absolute value
equation …
![Page 34: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/34.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 3 : Solve , and show the solution set as an interval. 532 x
![Page 35: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/35.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 3 : Solve , and show the solution set as an interval. 532 x
14
2
2
2
2
2
8
228
33 3
5325
x
x
x
x
![Page 36: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/36.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 3 : Solve , and show the solution set as an interval. 532 x
14
2
2
2
2
2
8
228
33 3
5325
x
x
x
x I like to graph the solution to determine the
interval…
4 1 xx
-1 4
![Page 37: Interval Notation and Absolute Value](https://reader031.fdocuments.net/reader031/viewer/2022020404/577cc4fb1a28aba7119ae775/html5/thumbnails/37.jpg)
CALCULUS 1 – Algebra review
Absolute Value Equations
Remember, absolute value equations have two possible answers;
positive and negative. So when solving, drop the absolute value
sign, and set the equation equal to the original answer, and also it’s
negative counterpart.
EXAMPLE # 3 : Solve , and show the solution set as an interval. 532 x
14
2
2
2
2
2
8
228
33 3
5325
x
x
x
x I like to graph the solution to determine the
interval…
4 1 xx
-1 4
)4,1(interval