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CALCULUS 1 – Algebra review
Intervals and Interval Notation
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.
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
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
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
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..
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 ( > )
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.
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.
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
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 interval753 x
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 interval753 x
4
123
55
753
x
x
x
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 interval753 x
4
123
55
753
x
x
x
4 graph
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 interval753 x
4
123
55
753
x
x
x
4
) ,4 (
graph
interval
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
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
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
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
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
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
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
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
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”
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
012070 2
0
TRUEFALSETRUE
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
012070 2
0
TRUEFALSETRUE
,34,interval
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.
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
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
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
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
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
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
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 …
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
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
142
2
2
2
2
8
228
33 3
5325
x
x
x
x
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
142
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
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
142
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