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Math 112 Precalculus Mathematics Sections 1 and 4 Instructor: Christopher Davis Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 1 / 19

Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

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Page 1: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Math 112 Precalculus MathematicsSections 1 and 4

Instructor: Christopher Davis

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 1 / 19

Page 2: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Who am I?

Christopher Davis,Office: Hibbard 533eMail: [email protected] hours:11AM - 12PM Monday, Tuesday, Wednesday, Thursday(also by appointment, but please give me warning)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 2 / 19

Page 3: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Syllabus, the course webpage and role:

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 3 / 19

Page 4: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Overview of this class

today:

A review: Real numbers, intervals, Absolute value.

soon:

Solving equations to get the information you want.

Functions and Graphs - Encoding an visualizing data.

Polynomials and their applications.I The easiest of functions to understand.I Applications to optimization (making the best possible choice).

Exponentials and LogarithmsI Population growth, radioactive decay and compound interest are all the

same thing.

Trigonometry: The mathematics of triangles and ballistics.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 4 / 19

Page 5: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Overview of this class

today:

A review: Real numbers, intervals, Absolute value.

soon:

Solving equations to get the information you want.

Functions and Graphs - Encoding an visualizing data.

Polynomials and their applications.I The easiest of functions to understand.I Applications to optimization (making the best possible choice).

Exponentials and LogarithmsI Population growth, radioactive decay and compound interest are all the

same thing.

Trigonometry: The mathematics of triangles and ballistics.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 4 / 19

Page 6: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Overview of this class

today:

A review: Real numbers, intervals, Absolute value.

soon:

Solving equations to get the information you want.

Functions and Graphs - Encoding an visualizing data.

Polynomials and their applications.I The easiest of functions to understand.I Applications to optimization (making the best possible choice).

Exponentials and LogarithmsI Population growth, radioactive decay and compound interest are all the

same thing.

Trigonometry: The mathematics of triangles and ballistics.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 4 / 19

Page 7: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Overview of this class

today:

A review: Real numbers, intervals, Absolute value.

soon:

Solving equations to get the information you want.

Functions and Graphs - Encoding an visualizing data.

Polynomials and their applications.I The easiest of functions to understand.I Applications to optimization (making the best possible choice).

Exponentials and LogarithmsI Population growth, radioactive decay and compound interest are all the

same thing.

Trigonometry: The mathematics of triangles and ballistics.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 4 / 19

Page 8: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:

7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 9: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 10: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is

any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 11: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 12: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .

−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 13: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 =

− .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 14: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666

You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 15: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 16: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0

1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 17: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1

−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 18: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1

2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 19: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 20: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 21: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 22: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Topic 1: Real numbers.

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 5 / 19

Page 23: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.

Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval?

yes

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 24: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval?

yes

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 25: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval?

yes

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 26: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval?

yes

Is 4 in this interval?

no

Is −3 in this interval?

It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 27: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval? yes

Is 4 in this interval?

no

Is −3 in this interval?

It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 28: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval? yes

Is 4 in this interval? no

Is −3 in this interval?

It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 29: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval? yes

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 30: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval? yes

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 31: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval? yes

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 6 / 19

Page 32: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

How many different things might we mean by the interval from −3 to 1?

We could include −3 and 1.

We could exclude −3 and 1.

We could include −3 and exclude 1.

We could exclude −3 and include 1.

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.Example:

0)1

(

−3

Excludes −3 and 1.

0]1

(

−3

Excludes −3 and includes 1.

Draw the interval which includes both endpoints.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 7 / 19

Page 33: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

How many different things might we mean by the interval from −3 to 1?

We could include −3 and 1.

We could exclude −3 and 1.

We could include −3 and exclude 1.

We could exclude −3 and include 1.

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.Example:

0)1

(

−3

Excludes −3 and 1.

0]1

(

−3

Excludes −3 and includes 1.

Draw the interval which includes both endpoints.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 7 / 19

Page 34: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

How many different things might we mean by the interval from −3 to 1?

We could include −3 and 1.

We could exclude −3 and 1.

We could include −3 and exclude 1.

We could exclude −3 and include 1.

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.

Example:

0)1

(

−3

Excludes −3 and 1.

0]1

(

−3

Excludes −3 and includes 1.

Draw the interval which includes both endpoints.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 7 / 19

Page 35: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

How many different things might we mean by the interval from −3 to 1?

We could include −3 and 1.

We could exclude −3 and 1.

We could include −3 and exclude 1.

We could exclude −3 and include 1.

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.Example:

0)1

(

−3

Excludes −3 and 1.

0]1

(

−3

Excludes −3 and includes 1.

Draw the interval which includes both endpoints.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 7 / 19

Page 36: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals

How many different things might we mean by the interval from −3 to 1?

We could include −3 and 1.

We could exclude −3 and 1.

We could include −3 and exclude 1.

We could exclude −3 and include 1.

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.Example:

0)1

(

−3

Excludes −3 and 1.

0]1

(

−3

Excludes −3 and includes 1.

Draw the interval which includes both endpoints.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 7 / 19

Page 37: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8].

True

−4 is in (−4, 8].

False

−4.25 is in (−4, 8].

False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

Page 38: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8].

True

−4 is in (−4, 8].

False

−4.25 is in (−4, 8].

False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

Page 39: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8].

True

−4 is in (−4, 8].

False

−4.25 is in (−4, 8].

False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

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Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8].

True

−4 is in (−4, 8].

False

−4.25 is in (−4, 8].

False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

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Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8]. True

−4 is in (−4, 8].

False

−4.25 is in (−4, 8].

False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

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Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8]. True

−4 is in (−4, 8]. False

−4.25 is in (−4, 8].

False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

Page 43: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8]. True

−4 is in (−4, 8]. False

−4.25 is in (−4, 8]. False

8 is in (−4, 8].

True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

Page 44: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Making notation easier.

Drawing a picture like:

0]1

(

−3

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8]. True

−4 is in (−4, 8]. False

−4.25 is in (−4, 8]. False

8 is in (−4, 8]. True.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 8 / 19

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A Bat-Example

After unraveling a puzzle, Batman knows that the Riddler is in the interval[3, 6]. After investigating, Batman knows he is not to be found in [4, 5].

Where could the Riddler still be hiding? Draw a picture and write downthe names of the intervals where the Riddler might be.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 9 / 19

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A Bat-Example

After unraveling a puzzle, Batman knows that the Riddler is in the interval[3, 6]. After investigating, Batman knows he is not to be found in [4, 5].

Where could the Riddler still be hiding? Draw a picture and write downthe names of the intervals where the Riddler might be.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 9 / 19

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Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define? Answer: [3, 5).

What inequalities define (4, 6)? Answer: 4 < x < 6 (or 6 > x > 4)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 10 / 19

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Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define? Answer: [3, 5).

What inequalities define (4, 6)? Answer: 4 < x < 6 (or 6 > x > 4)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 10 / 19

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Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define?

Answer: [3, 5).

What inequalities define (4, 6)? Answer: 4 < x < 6 (or 6 > x > 4)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 10 / 19

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Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define? Answer: [3, 5).

What inequalities define (4, 6)? Answer: 4 < x < 6 (or 6 > x > 4)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 10 / 19

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Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define? Answer: [3, 5).

What inequalities define (4, 6)?

Answer: 4 < x < 6 (or 6 > x > 4)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 10 / 19

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Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define? Answer: [3, 5).

What inequalities define (4, 6)? Answer: 4 < x < 6 (or 6 > x > 4)

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 10 / 19

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Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[

0−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

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Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[

0−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

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Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[0

−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

Page 56: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[0

−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

Page 57: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[0

−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

Page 58: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[0

−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

Page 59: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

[0

−2

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 11 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] [a,∞)

[a, b) (−∞, b)

(a, b] (−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] a ≤ x ≤ b [a,∞)

[a, b) (−∞, b)

(a, b] (−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) (−∞, b)

(a, b] (−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) a ≤ x < b (−∞, b)

(a, b] (−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] (−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] a < x ≤ b (−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞)

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞)

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞) a ≤ x

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

Page 70: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞) a ≤ x[

a

[a, b) a ≤ x < b[

a

)b

(−∞, b)

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞) a ≤ x[

a

[a, b) a ≤ x < b[

a

)b

(−∞, b) x < b

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞) a ≤ x[

a

[a, b) a ≤ x < b[

a

)b

(−∞, b) x < b)b

(a, b] a < x ≤ b(

a

]b

(−∞, b]

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞) a ≤ x[

a

[a, b) a ≤ x < b[

a

)b

(−∞, b) x < b)b

(a, b] a < x ≤ b(

a

]b

(−∞, b] x ≤ b

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

a

)b

(a,∞) a < x(

a

[a, b] a ≤ x ≤ b[

a

]b

[a,∞) a ≤ x[

a

[a, b) a ≤ x < b[

a

)b

(−∞, b) x < b)b

(a, b] a < x ≤ b(

a

]b

(−∞, b] x ≤ b]b

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 12 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| =

5

Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?

0 5

distance is 5

|5| =

5

Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| =

5

Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| =

5

Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| =

5

Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| =

5Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| = 5

Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| = 5Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| = 5Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| = 5Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5

Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| = 5Example: What is | − 5|?

0−5

distance is 5

| − 5| =

5Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

|5| = 5Example: What is | − 5|?

0−5

distance is 5

| − 5| = 5Evaluate |4− 3|?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 13 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | =

x .

When x is negative, |x | =

− x .

|0| =

0

Compute:

5− |1− 7|

=

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

=

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

=

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | =

− x .

|0| =

0

Compute:

5− |1− 7|

=

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

=

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

=

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| =

0

Compute:

5− |1− 7|

=

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

=

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

=

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:

5− |1− 7|

=

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

=

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

=

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| =

5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| =

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| =

5− 6 = −1

||2− 3| − | − 4|| =

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| =

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| =

|1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| =

| − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| =

3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

=

4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.Since x > 4, 4− x < 4− 4 = 0.

4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.

|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | = x .

When x is negative, |x | = − x .

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 14 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean?

Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) =

x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 =

5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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A harder example

When x is positive, |x | = x .

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 15 / 19

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The distance between numbers

What is the distance from −3 to 2?

−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

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The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

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The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

Page 117: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

Page 118: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3?

Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

Page 119: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

Page 120: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

Page 121: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

The distance between numbers

What is the distance from −3 to 2?−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 16 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2

[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2

[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2

[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this?

Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

Page 129: Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review: Real numbers, intervals, Absolute value. soon: Solving equations to get the information

Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0

−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2[ ]

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

0−2 2

()

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 17 / 19

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As a table

Complete this tableinequality withabsolute value

interval(s)inequality without

absolute value

|y | < 2

y is in (−2, 2)

)( −2 < y < 2

|y | ≤ 2y is in [−2, 2]

][

−2 ≤ y ≤ 2

|y | > 2y is in (2,∞) or (−∞, 2)

()

y < −2 or 2 < y

|y | ≥ 2

y is in [2,∞) or (−∞, 2]

() y ≤ −2 or 2 ≤ y

What should this table look like if 2 is replaced with any positive numberd?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 18 / 19

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As a table

Complete this tableinequality withabsolute value

interval(s)inequality without

absolute value

|y | < 2

y is in (−2, 2)

)( −2 < y < 2

|y | ≤ 2y is in [−2, 2]

][ −2 ≤ y ≤ 2

|y | > 2y is in (2,∞) or (−∞, 2)

() y < −2 or 2 < y

|y | ≥ 2

y is in [2,∞) or (−∞, 2]

() y ≤ −2 or 2 ≤ y

What should this table look like if 2 is replaced with any positive numberd?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 18 / 19

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As a table

Complete this tableinequality withabsolute value

interval(s)inequality without

absolute value

|y | < 2y is in (−2, 2)

)( −2 < y < 2

|y | ≤ 2y is in [−2, 2]

][ −2 ≤ y ≤ 2

|y | > 2y is in (2,∞) or (−∞, 2)

() y < −2 or 2 < y

|y | ≥ 2y is in [2,∞) or (−∞, 2]

() y ≤ −2 or 2 ≤ y

What should this table look like if 2 is replaced with any positive numberd?

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 18 / 19

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One last problem

What is the set of all x such that |x − 5| ≤ 4?Does it include any negative numbers?Your group must check your answer with me before leaving.

Homework: Appendix A.1: 39, 40, 49, 50, 57, 58; Appendix A.10: 1, 2,3, 4, 10, 11, 12, 13, , 14, 23, 24, 25, 26.

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 19 / 19