math 150 notes unit 1 fall 16 - texas a&m university 150 notes! section 1a ! page 1 4 5, −17...

Chapter 1A -- Real NumbersMath Symbols:

∈ ∀

∃ ∍iff ⇔ or↔

∪ ∩

Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16, ...} and let B = {3, 6, 9, 12, 15, 18, 21, 24, 30, ...}

Then A ∪B= _______________________________________________________________

and A ∩B= ________________________________________________________________

Sets of Numbers

Name(s)fortheset Symbol(s)fortheset

1,2,3,4,… NaturalNumbersPositiveIntegers

…,-3,-2,-1 Negativeintegers

0,1,2,3,4,… Non-negativeIntegersWholeNumbers

…,-3,-2,-1,0,1,2,3,… Integers

Note: _________________________________ is the German word for number.

©Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A !

45

, −1713

, 129 Rational Numbers

Note: This is the first letter of ____________________________

1. Circle One:

a) 5 is a rational number! ! ! TRUE!! FALSE

b) π4

is a rational number!! ! TRUE!! FALSE

c) 165

is a rational number! ! TRUE!! FALSE

d) 175

is a rational number! ! TRUE!! FALSE

e) 0 is a rational number!! ! TRUE!! FALSE

Definition of a rational number: If a, b ∈! and b ≠ 0 , then ab ∈! .

Translation: _________________________________________________________________

___________________________________________________________________________

Decimal Expansions of Rational Numbers: Fact: The decimal expansions of rational numbers either

__________________________ or __________________________

Examples ! 14

= ________________________ ( This is a terminating decimal.)

49

= _____________________ (This is a repeating decimal.)


You know how to express a terminating decimal as a fraction. For example .25 = 25100

, but how

do you express a repeating decimal as a fraction?

2. Express these repeating decimals as the quotient of two integers:

a) .373737…or .37

Let x = .37373737...

Then100x =

b) .123

Let x = .123123123...

Then

There are a lot of rational numbers, but there are even more irrational numbers. Irrational

numbers cannot be expressed as the quotient of two integers. Their decimal expansions

never terminate nor do they repeat.

Examples of irrational numbers are _____________________________________________

The union of the rational numbers and the irrational numbers is called the

_____________________________ or simply the _____________. It is denoted by _______.



4.W

hatpropertiesarebeingused?

Nam

e of

Pro

perty

a) (−40)+8(x+5)

=8(x+5)+(−40)

a+b=b+a

∀

a,b

∈!

b)8(x+5)+(−40)=8x

+40

+(−40)

a(b+c)=ab

+ac

∀

a,b

,c∈!

c)8x

+40

+(−40)=8x

+(40+(−40))

(a+b)

+c=

a+

(b+c)

∀

a,b

,c∈!

d)0+5=5+0=5

0+a=a+0=a

∀

a∈!

e)7+(−7)

=(−7)

+7=0

For e

ach a∈!, ∃

an

othe

r ele

men

t of

!de

note

d by

−a

suc

h th

at

a+(−a)

=(−a)

+a=0

f)6⋅1=1⋅6=6

1⋅a=a⋅1=

a

∀ a

∈!

g)4⋅

1 4⎛ ⎝⎜⎞ ⎠⎟=1 4⎛ ⎝⎜⎞ ⎠⎟⋅4=1

For e

ach a∈!, a≠0

∃

anot

her

elem

ent o

f !de

note

d by

1 a s

uch

that

a⋅1 a⎛ ⎝⎜⎞ ⎠⎟=1 a⎛ ⎝⎜⎞ ⎠⎟⋅a=1

Properties of Real Numbers

Notice that the commutative property considers the ________________ of the numbers.

When demonstrating the associative property, the order of the numbers stays the same

but the ____________________ changes.

When do we use the commutative property of addition? Consider 7 + 28 + 43 .!The rules for order of operation tell us to add left to right, but the commutative property of

addition allows us to change the order to make our life easier:

7 + 29 + 43= ______________________________________________

When do we use the associative property of multiplication? Consider 16 × 25 .

16 × 25 = ___________________________________________________________________

Is subtraction commutative? ___________ Example: _____________________________

Is division commutative? ___________ Example: _____________________________

Consider 12 − 3( )− 4 = __________________ while 12 − 3− 4( )= ___________________

So is subtraction associative? ___________

Consider 12 ÷ 3( ) ÷ 4 = __________________ while 12 ÷ 3÷ 4( )= ___________________

So is division associative? ___________


Notice that there are 2 identities:

The additive identity is ______________. ! The multiplicative identity is ___________.

Notice there are 2 kinds of inverses.

The additive inverse of 4 is ____________. Note that _____________________________

The additive inverse of -9 is ____________________. Note that ______________________

What is true about the sum of a number and its additive inverse? ______________________

Every real number has an additive inverse.

In rigorous settings, subtraction is defined as addition with the additive inverse. In other words

5 − 2 = 5 + −2( ) or more generally a − b = a + −b( )

The multiplicative inverse of 4 is ________. Note that _______________________________

The multiplicative inverse of -9 is ______________ Note that ________________________

What is true about the product of a number and its multiplicative inverse? ________________

Almost all of the real numbers have a multiplicative inverse.

Which real number does NOT have a multiplicative inverse? _________________________


Absolute Value

4 = _______________ −9 = _______________

Formal definition for absolute value

! ! ! ! x =⎧⎨⎪

⎩⎪

It will help if you learn to think of absolute value as _________________________________

(Notice that -4 is 4 units away from the origin.)

Distance between two points:

3.) Find the distance between

a) 4 and 1

b) -1 and 5

c) 4 and -2

d) -3 and -4

e) 1 and x

f) x and y


Notice that! 4 −1 = 1− 4 ! and ! −3− 2 = 2 − (−3) . In general x − y = ________

Notice that! −5 + 4 = _____ ! whereas −5 + 4 = _____ So, in general x + y ≠ _______

Understanding Zero

True or False :! ! 06= 0 !

True or False :! ! 60= 0 ! !

True or False :! ! 00= 0 ! !

For a, b∈! , ab= 0 ⇔ __________________ and ___________________

Another way to say this: a fraction is zero when its _____________________ is zero and its

____________________ is not.

This concept will be very important when we solve rational equations like 1x +1

= x + 23− x

Notation inconsistency:

When we write 4x it means the product of 4 and x .

When we write 4 13 it means the sum of 4 and

13

4 13 is usually called a mixed number whereas the corresponding ____________ is termed an

improper fraction. There is nothing improper about an improper fraction.

Extra Problems: Text: 1-9, 22, 23


Chapter 1B - Exponents and RadicalsMultiplication is shorthand notation for repeated addition 4 + 4 + 4 + 4 + 4 = 5 × 4

Exponents are shorthand notation for repeated multiplication. So

2 × 2 × 2 × 2 × 2 = _______ xixixix = __________

You probably know that 3−1 = _________ and x−5 = ____________ this is because

by definition x−1 = __________ . In general if m ∈ Ν, then x−m =__________

Properties of exponents

Example In general

x4x3 xmxn

x8

x5xm

xn

x3

x3x0 for x ≠ 0

Note: 00 ≠ 0 and 00 ≠1 00 is an _________________________________

More Properties of Exponents

Example In general

x2( )3

(xy)3 (xy)m

xy

⎛

⎝⎜

⎞

⎠⎟2

xy

⎛

⎝⎜

⎞

⎠⎟m

©Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B !

1. Simplify

a) (−8)2 b) −82 c) 30

d) 12−6

⎛⎝⎜

⎞⎠⎟ e) 1

x−2⎛⎝⎜

⎞⎠⎟

!

e) x−3

4 f) 3x

⎛⎝⎜

⎞⎠⎟−1

g) 25x5y−2

15x2y−1⎛⎝⎜

⎞⎠⎟

3 h) 12x−3y2

4x4y−1⎛⎝⎜

⎞⎠⎟

−2


Roots or Radicals

TRUE or FALSE: 4 = ±2 .

Important distinction: x2 = 25 has ______ solutions: x = _______ and x = _____

25 represents exactly _________ number. 25 = ________.

Recall that 25 = 32 . Now suppose you have x5 = 32 with the instructions: “Solve for x .”

How do you express x in terms of 32? The vocabulary word is __________________

In this case x equals the _______________________________.

The notation is x = ____________________ or x = ____________________

This second notation is called a fractional exponent.

Examples: ! 814 = 81( )14

⎛⎝⎜

⎞⎠⎟ = ____________ because 3 = _________.

! ! ! 125( )13

⎛⎝⎜

⎞⎠⎟ = ____________ because 5 = _________.

Properties of Roots (just like properties of exponents!)

Example In general

27i83

273 83

xym


Example In general

364

364

⎛

⎝⎜

⎞

⎠⎟

xm

ym

Now consider 823 . Notice that 82( )

13 = _______________________________

Also 813

⎛⎝⎜

⎞⎠⎟⎛

⎝⎜⎞

⎠⎟

2

= _____________________________________________________

So it would seem that 823 = ________________________________________

In fact, more generally, it is true that xmn

⎛⎝⎜

⎞⎠⎟ = __________________________

Similarly, consider 643 = 6412

⎛⎝⎜

⎞⎠⎟⎛

⎝⎜⎞

⎠⎟

13

= _______________________________

Whereas, 643 = 6413

⎛⎝⎜

⎞⎠⎟⎛

⎝⎜⎞

⎠⎟

12

= _____________________________________

So it would seem that 643 _______________________________________

In general xnm _______________________________________________


Simplify xnn for all n∈! .

On Worksheet 1 you graphed y = x33 . Hopefully you found that its graph looked just like the

graph of y = x . In other words, you saw that x33 = ____________

You also graphed y = x22 . Hopefully you found that its graph looked just like the graph of

y = x . In other words, you saw that x22 = ____________

Most people believe that 255 = _______ and agree easily that 2nn = _______ ∀ n

But −2( )44 = _______________________ while −2( )55 = _______________________

So hopefully you see believe −2( )nn = 2 whenever n is ___________

and −2( )nn = −2 whenever n is ___________

So to simplify xnn for any x ∈! and any n∈! we have

xnn =⎧⎨⎪

⎩⎪

When simplifying, we don’t want to use absolute value signs if they are unnecessary.

Obviously we would write 4 = 4 . But also x4 = ____________________________

_______________________________________________________________________

Also, for the time being, if n is even, xn is defined only if x > 0 . In other words, for the time

being −4 and −174 are not defined. So x3 is only defined if x > 0 . That means that

x3 = _______________________________________________


Simplifying Radicals A radical expression is simplified when the following conditions hold: 1. All possible factors (“perfect roots”) have been removed from the radical. 2. The index of the radical is as small as possible. 3. No radicals appear in the denominator.

2. Simplify

a) −325

b) (−216)13

⎛⎝⎜

⎞⎠⎟

c) −912

⎛⎝⎜

⎞⎠⎟ d) −64

e) (−13)66 f) 16x44

g) x53 h) 168

i) −12564

⎛⎝⎜

⎞⎠⎟

−23


Rationalizing the Denominator

“Rationalizing the denominator” is the term given to the techniques used for eliminating radicals from the denominator of an expression without changing the value of the expression. It involves multiplying the expression by a 1 in a “helpful” form.

3. Simplify

a) 12! ! ! ! ! b) 1

23

c) 14x3

! ! ! ! ! ! d) 16x−2

27y4

Notice: x + 3( ) x − 3( ) = __________________________________________.

To simplify 12 − 5

we multiply it by 1 in the form of


So 12 − 5

=

and 9x + 3 5

=


Adding and Subtracting Radical Expressions

Terms must be alike to combine them with addition or subtraction. Radical terms are alike if they have the same index and the same radicand. (The radicand is the expression under the radical sign.)

4. Simplify

a) 5 + 2 7 − 3 5 − 7 b) 128x43 − 3x 18x − x 2x3( )

Please notice 9 + 16 = ______________________! Whereas 25 = ___________

In other words ____________________________

In general __________________________

True or False: 2x + 3x = 5x

True or False: 3x + 3x = 6x

3x + 3x = ______________________________


Interesting cases of xnn when n is even

For the time being if n is even, xn is undefined if x < 0 in other words −26 is undefined.

Moreover, we want to remember that when x > 0 , x = x ; we always simplify 4 = ______

To simplify the following, start by considering for what values of x the expression is defined

x44 Here the index of the radical is even, but the expression under the radical is always

positive, so the expression is defined for all values of x x44= _____________________

x54 Here the index of the radical is even, but x5 could be negative if x was negative.

We have to recognize that x54 is only defined if x > 0 so x54

= ________________

x84 Here the index of the radical is even, and the expression under the radical is always

positive, so the expression is defined for all values of x x84= _____________________

x124Here the index of the radical is even, and the expression under the radical is always

positive, so the expression is defined for all values of x x124= ____________________

Life is so much easier when the index of the radical is odd, because when the index of the

radical is odd, the expression is always defined

x33= _________!! ! −15 = _________!! ! −325 = _________

Extra Problems: Text: 1-32


Chapter 1C - PolynomialsWhich of these are examples of polynomials?

Polynomial Not a polynomial Polynomials do not have

x2 + 2x +1

x7 + π

2( ) x2 + 2x +14 x −1

1x+ 3x2

5x3 + 3x2 − 2( )14

5x3 + 3x2 − 2( )2

x2 + sin x

5

Polynomials are expressions of the form anxn + an−1x

n−1 +!+ a1x + a0 where ai ∈! and n∈!

n is called ___________________________________________

an is called ___________________________________________

anxn is called ___________________________________________

a0 is called ___________________________________________

If n = 2, the polynomial is called a ___________________________ Example:

If n = 3, the polynomial is called a ____________________________ Example:

A polynomial with 2 terms is called a ___________________________ Example:

A polynomial with 3 terms is called a ___________________________ Example:

©Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C !

Factoring Polynomials Factoring polynomials will help us in later chapters when we need to find the roots of polynomials. Notice that we are working with expressions, not equations. When the instructions say, “Factor,” you will be given an expression and asked to change its form so that it is written as a product rather than a sum.

Common Factors: 12x4 + 4x2 + 2x ____________________________________

Factor by Grouping: (Factoring by grouping is your best bet if you have a polynomial with four terms!)

3x3 + x2 −12x − 4 ___________________________________________________________________________

Factor quadratics using special patterns:

Look what happens when you multiply and simplify:

(x − 3)(x + 3) _________________________________

(4 − y)(4 + y) __________________________________

Difference of Squares a2 − b2 =

Look what happens when you expand

(x + 5)2 __________________________________________________

(x − 2)2 __________________________________________________

Square of a Binomial a2 + 2ab + b2 =

a2 − 2ab + b2 =

1. Factor

a) x2 −144 b) 16x2 − 81

c) x2 +10x + 25 d) 16x2 + 8x +1


Factoring quadratics, lead coefficient 1:

To factor x2 +10x +16 , determine if there are 2 numbers whose product is _______ and

whose sum is ___________. Those two numbers are ______ and ______ so

x2 +10x +16 = ___________________________________________________________

To factor x2 − 2x − 48 , determine if there are 2 numbers whose product is _______ and

whose sum is ___________. Those two numbers are ______ and ______ so

x2 − 2x − 48 = __________________________________________________________

Factoring quadratics with lead coefficient is not 1

Use the Blankety-Blank Method!

To factor: 10x2 +11x + 3 determine if there are 2 numbers whose product is _______

and whose sum is ___________ Those two numbers are _________ and ________

Now rewrite 10x2 +11x + 3 =___________________________________________________

To factor: 3x2 −14x − 5 determine if there are 2 numbers whose product is _______

and whose sum is ___________ Those two numbers are _________ and ________

Now rewrite 3x2 −14x − 5 =____________________________________________________

Important Fact: a2 + b2 does not factor over the real numbers. So x2 + 4 cannot be factored

using real numbers.


Factoring polynomials in quadratic form

2. Factor completely (Sometimes factors need to be factored.)

a) 9x4 − 36 b) x6 − 5x3 − 6

c) x10 + 2x5 +1 4x12 − 9x6 + 2

Factoring special cubics

Look what happens when you multiply

(a + b)(a2 − ab + b2 )

Similarly look what happens when you multiply

(a − b)(a2 + ab + b2 )


Sum of Cubes a3 + b3

Difference of Cubes a3 − b3

3. Factor completely (Sometimes factors need to be factored.)

a) 27 − x3 b) 64x3 +1

c) x6 + 8 d) x5 − 4x3 − x2 + 4

e) 3x3 + 6x2 −12x − 24 f) 16x4 − 25x2


Dividing Polynomials (This is a lot like long division of integers!)

4. 4x3 − x2 − 5x + 2

x −1

What is the dividend? ___________________________ What is the divisor? ____________

What is the quotient? _______________________________


5. 3x3 − 2x2 + 5x2 −1

= _________________________________________!

What is the dividend? ___________________________ What is the divisor? ____________

What is the quotient? _______________________________

Extra Problems: Text: 1, 2, 16, 17, 18, 21-38


Chapter 1D - Rational ExpressionsA rational expression is the quotient of two _________________________________

In this section we’ll add, subtract, multiply, and divide rational expressions. Our goal will be to

simplify the result so that it is expressed as

the quotient of two factored polynomials.

We will also need to be aware of possible restrictions on the values of our independent

variable(s), usually x .

Rational expressions are undefined when _____________________________________

So the rational expression xx

= ___________ as long as _____________________

Sometimes we will simplify a rational expression by canceling common factors.

This is similar to reducing fractions 1518

= 33

⎛⎝⎜

⎞⎠⎟56

⎛⎝⎜

⎞⎠⎟ =

56

Notice that 1518

= 3+126 +12

, but we would not cancel the 12’s!

When simplifying rational expressions we look for common ______________________ in the

numerator and denominator.

Simplifying Rational Expressions:

1. Simplify the rational expressions, if possible, and list any restrictions that exist for x .

a) x −1x

! ! ! ! ! ! Restrictions on x : _____________________

©Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1D !

b) x + 2x2 + 4

! ! ! ! ! ! Restrictions on x : _____________________

c) x7 − x4

x 5−x3! ! ! ! ! ! Restrictions on x : ______________________

Addition and Subtraction of Rational Expressions

2. Simplify

a) 3x + 2

+ 2x − 3

! ! ! ! ! Restrictions on x : _____________________

b) x − 4x +1

− x2 − 8x +16x2 −1

! ! ! ! Restrictions on x : _____________________


Multiplication and Division of Rational Expressions

3. Simplify

a)

8x + 2

⎛⎝⎜

⎞⎠⎟ i

x2 − x − 2x2 −1

⎛⎝⎜

⎞⎠⎟

! ! ! ! Restrictions on x : ______________________

b) x2 − 4x − 21x + 4

÷ (x2 + 7x +12) ! ! ! Restrictions on x : ______________________

c) x3 −1x +1x −1

x2 + 2x +1

=! ! ! ! ! Restrictions on x : _____________________


Compound FractionsA compound (or complex) fraction is an expression containing fractions within the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division.

4. Simplify

a) 3x+ 12x − 75x+1

! ! ! ! ! Restrictions on x : _____________________

b) 3 x + h( )−2 − 3x−2h

=

Restrictions on x : ____________________ Restrictions on h : ____________________



Chapter 1E - Complex Numbers

−16 exists! So far the largest (most inclusive) number set we have discussed and the one we have the most experience with has been named the real numbers.

And ∀ x ∈ ! , x2 ≥ 0

But there exists a number (that is not an element of ! ) named i and i2 = _____

Since i2 =_____, −1 = __________ , so −16 = ________________________

i is not used in ordinary life, and humankind existed for 1000’s of years without considering i .

i is, however, a legitimate number. i , products of i , and numbers like 2 + i are solutions to many problems in engineering. So it is unfortunate that it was termed “imaginary!”

i and numbers like 4i and −3i are called _____________________________

Numbers like 2 + i and 173− 5i are called ____________________________

More formally ! is the set of all numbers _______________________________

When a complex number has been simplified into this

______________ form, it is called _________________________________

a is called the ______________________ part. b is called the imaginary part.

So for the complex number 4 − 3i , ____ is the real part while _______ is the imaginary part.

1. Put the following complex numbers into standard form:

a) 16 + −9( ) + −13− −100( ) = ______________________________________________

b) 1+ 2i( )− (3− 4i) = ______________________________________________________

c) 5 + 4i( )(3− 2i) = _______________________________________________________

©Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1E !

d) i3 = ____________ e) i4 = ____________ f) i5= ____________

Graphing Complex Numbers

When we graph elements of ! , we use ________________________________

When we graph elements of ! , we use the complex plane which will seem a lot like the Cartesian plane.

In the Cartesian plane, a point represents __________________________________

In the complex plane, a point represents ___________________________________

The horizontal axis is the ________________________________________________________________________

The vertical axis is the __________________________________________________________________________

2. Graph and label the following points on the complex plane:

A 3+ 4i ! ! ! B −2 + i

C 32− 3i ! ! ! D 3i

E −4 ! ! ! F −1− 2i


Absolute Value of a Complex Number:

If z ∈ ! then z is defined as its

_______________________________________

Calculate 3+ 4i (How far is 3+ 4i from the origin?)

3+ 4i = _____________

Consider a + bi , an arbitrary element of ! . ! !

What is a + bi ? (How far is a + bi from the origin?)

In general z = _____________________________

Notice that z is a real number. It is z ’s distance from the origin. We did not use i to

calculate z .

3. Calculate −5 + i . (How far is −5 + i from the origin?)

−5 + i = ___________________


Distance Between Two Complex Numbers

Plot and label two points in the complex plane z1 = 3+ 5i and z2 =1+ 2iThe distance between z1 and z2 is

Just for fun, calculate z1 − z2 = ___________________________________________

And z1 − z2 = ________________________________________________________

What we have seen is that for this particular z1 and z2 , the distance between these points is

equal to __________________. But this is actually true for all complex numbers.

The distance between two general points z1 and z2 is _______________

Complex Conjugate

If z1 = 3+ 4i , then z1= ________________ .

If z2 = −1− 2i , then z2 = __________ .

Graphically the complex conjugate is the

________________________ of the number through the

________________________

More generally if z = a + bi , then z = ______________________


4. Put the following complex numbers into standard form.

a) 2 − 3i = ________________! ! ! b) 4i = ___________________

c) 5 = ____________________! ! ! d) 3i − 2 = ___________________

Finally notice that 3+ 4i( ) 3− 4i( ) = ______________________________________

More generally a + bi( ) a − bi( ) = ________________________________________

in other words ziz = ________________________________________________

Dividing by complex numbers

5. 25 + 50i3+ 4i



Chapter 2A Solving EquationsThe solution of an equation is a value (or a set of values) that yields a true statement when substituted for the variable in an equation.

The word “solve” means determine ______________________________________

Solving Linear Equations

Linear equations are equations involving only polynomials of degree one.

Examples include 2x +1= −4 and 4x + 2 = −x − 3

The algebraic techniques to find the solutions to these equations are simple, but I want you to keep in mind that there is a geometric interpretation associated with the equation. We are looking for the intersection of two linear functions.

Here are the graphs of ! ! ! ! ! ! Here are the graphs of

______________ and! ! ! ! ! ! ______________ and

______________ ! ! ! ! ! ! ! ______________

See how the lines intersect at ! ! ! ! ! See how the lines intersect at

x =______________! ! ! ! ! ! ! x =______________

See how substituting x =________! ! ! ! See how substituting x =_______

into 2x +1= −4 ! ! ! ! ! ! ! ! into 4x + 2 = −x − 3

makes the statement true.! ! ! ! ! ! makes the statement true.


Solving Quadratic Equations

Quadratic equations are equations involving only polynomials of degree two.

Examples include x2 − x − 6 = 2x − 2 ! and! 5x2 − 5x = 2x2 − 3x +1

Geometrically, such equations could represent the intersection of a parabola and a line or the intersection of two parabolas.

x2 − x − 6 = 2x − 2 5x2 − 5x = 2x2 − 3x +112

⎛⎝⎜

⎞⎠⎟ x

2 = x −1


Solve Quadratic Equations by Factoring

Another technique for solving quadratic equations is by factoring. This technique is based on

the Zero-Product Principle. Think about it this way:

If a, b∈! and if ab =1 , must either a =1 or b =1 ? _____________________________

If ab = 0 , must either a = 0 or b = 0 ? _________________________________________

1. Solve a) 1− 2x2 = −x

Put in standard formax2 + bx + c = 0

Factor

Set each factor equal to zero

b) 6x2 = x +15

Put in standard formax2 + bx + c = 0

Factor

Set each factor equal to zero


Solve by completing the square.

This is an important technique that we will use in other types of problems throughout the semester. There will be a problem on exam 1 in which you will be asked to solve a quadratic equation by completing the square. Students who solve the equation correctly with methods other than completing the square will not receive credit for their solution.

Before we do this, let’s make some observations and practice some skills.

Expand the following expressions, looking for patterns:

(x − 3)2 = x2 ____ x ____

(x + 4)2 = x2 ____ x ____

(x − 5)2 = x2 ____ x ____

Now work backward...

(x _____)2 = x2 +16x ____

(x _____)2 = x2 + 22x ____

(x _____)2 = x2 −18x ____


2. Solve the following quadratic equations by completing the square:

a) x2 = 36 b) x − 2( )2 = 25

c) x2 + 2x +1= 16 d) x2 + 6x = 1

e) x2 + 9x +1= 0 f) 2x2 + x − 8 = 0


g) ax2 + bx + c = 0

If ax2 + bx + c = 0 , then x = _________________________________

This is called ________________________________________________

3. Solve using the quadratic formula: 2x2 = 6x − 3


Equations in quadratic form

4. Solve

a) x4 − 5x2 − 6 = 0 b) x10 + 2x5 +1= 0

c) 4x12 − 9x6 + 2 = 0


d) x43

⎛⎝⎜

⎞⎠⎟ − 5x

23

⎛⎝⎜

⎞⎠⎟ + 4 = 0


Solving Rational EquationsRational equations are equations involving rational functions. Solving rational equations is equivalent to finding the intersection(s) between 2 rational functions. Algebraically what we do is to “set one side of the equation equal to zero” and remember that a fraction is zero when

___________________________________________________________________

The mistake: Consider solving 2x= 3x

. A lot of people want to start by cross multiplying or

multiplying both sides of the equation by x. This would yield ________________________.

But neither 2

nor 3

are defined. This equation does not have a solution. The risk occurs

when multiplying both sides of an equation by something that could be zero.

5. Solve

a) 12x

= 2 + 15x


b) 4x − 4

− 3x −1

= 1


Radical Equations

To Solve Radical equations:I. Isolate one radical.II. Raise both sides of the equation to the appropriate power to remove the radical.! (Usually this means square both sides of the equation.)III. Repeat the process until all radicals have been removedIV. Check for extraneous solutions!

Why is it important to check for extraneous solutions to equations involving radicals?

How many solutions exist for the equation x = 3 ? __________________

Square both sides of the equation: ___________________________

What are the solutions to this equation?__________________________

See how ________________ is not a solution to the original equation?!

REMEMBER: (a + b)2 ≠ a2 + b2 (a + b)2 = _______________________

6. Solvea) 3 = x + 2x − 3


b) 8x +17 − 2x + 8 = 3

c) x + x − 5 = 1


Equations with Absolute Values

! ! Recall the definition of x =

7. Solve

a) x = 4

b) x − 3 = 2

c) 8 − x + 2 = 7


d) x +1 + 3= 2

e) x2 − x −12 = 8

! !


Equations with several variables

Equations in science and engineering often include many variables, and it is useful to be able to solve for one of the variables in terms of the others.

! ! Warm up: If 1x= 12+ 13

, then what is the value of x ?

8. The following equation comes from the physics of circuits 1Req

= 1R1

+ 1R2

+ 1R3

Solve for Req .


9. Ts = 2πmk

Solve for k .

10. Solve for y x = 3y2y − 5



Chapter 2B - Solving Inequalities

True or False: 4 < 4 ! ! ! True or False: 4 ≤ 4

Also though 3 < 4 , -4 < -3 which could also be written -3 > -4.

Written more generally, if a < b , then _____________________

This leads to the “rule” that when you multiply both sides of an inequality by a negative

number, you change the direction of the inequality.

A solution set is the set of values that make a (mathematical) statement true.

Absolute Value Inequalities

Determine the solution sets for the following inequalities

Number Line Inequality Interval Notation

x ≤ 3

x > 3

x > −3

x < −3

This kind of bracket ______________ indicates that the endpoint of the interval is included.

This kind of parenthesis __________ means that the endpoint of the interval is not included.

The interval notation for this set −3≤ x < 3 is __________________________! !


1. Solve

a) x < 4

b) x ≥ 2

c) x + 2 <1


d) 2x − 4 ≥ 3

e) 5 − 2x > 4


f) 4 − x − 3 >1


2. Think about solving the following inequalities graphically.

a) x + 3> 0 b) x + 3( ) 2 − x( ) > 0

Where does x + 3= 0 ? At x = ___________

Graphing the point (3, 0) divides the x-axis

into 2 regions.

When x < -3 the graph is _______________

the x-axis so x+3 _____ 0

When x > -3 the graph is _______________

the x-axis so x+3 _________ 0

x+3 changes sign at x=-3. x+3 goes from

negative to positive.

Where does x + 3( ) 2 − x( ) = 0 ?

At x = ___________ and x = ___________

These zeros divide the x-axis into 3 regions.

Notice that when x < -3 , the graph is below

the x-axis. Also notice how one factor is

negative, but the other factor is positive:

x+3 < 0 , but 2-x > 0. That means their

product will be negative when x < -3.

When -3 < x < 2, the graph is above the x-

axis. Look at the factors. Both factors are

positive: x+3 > 0 and 2-x > 0 . That means

their product will be positive when -3 < x < 2

Finally when x > 2, the graph is below the x-

axis. This corresponds to one factor being

positive while the other is negative

x+3 > 0 but 2-x < 0. So product of these two

factors is negative.


Nonlinear Inequalities

To solve nonlinear inequalities:

I. Move every term to one side of the inequality by adding or subtracting .

II. Factor the nonzero side.

III. Determine the critical values. (Critical values are those that make the expression zero or undefined. More specifically the values that make the numerator and/or denominator zero.)

IV. Let the critical numbers divide the number line (x-axis) into intervals.

V. Determine the sign of each factor in each interval.

VI. Use the sign of each factor to determine the sign of the entire product or quotient.

3. Solve

a) (x +1)(x − 2) > 0


b) x2 − x ≤ 6

TRUE or FALSE: 2 < 3

TRUE or FALSE: 2x < 3x

because when x < 0 , ______________

So don’t multiply both sides of an inequality by a variable that could be either positive or negative.

c) 2x<1


d) − 4x +1

> 12 − x


Linear Inequalities

4. Solve

a) 1− x > x + 3

b) 1− 2x <17 − 4x ≤ 8 − x


c) 2x + 4 < x + 2 < x2+ 3

d) 4x +1≤ 4 − 2x ≤ 3− x



math 150 notes unit 1 fall 16 - texas a&m university 150 notes! section 1a ! page 1 4 5, −17...

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