interval notation and absolute value

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 interval 753 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 interval 753 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 interval 753 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 interval 753 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

0120702

0

TRUE FALSE TRUE

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

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

14

2

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

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

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