Math Exam Review Semester 2 By Kyle Skarr and Ryan McLaughlin

Math Exam Review Semester 2

By Kyle Skarr and Ryan McLaughlin

Solving First Power Equations in one Variable

Example problem 4x=24-2x

How to solve 4x=24-2x +2x +2x

6x=24 /6 /6 X=4

Solving First Power Equations in one Variable continued

Equations containing fraction coefficients

– Example equation

3

5 4 2

x x

Least common denominator is 20

4 15 10x x

6 15 2.5x x


Equations with variables in the denominators-

– Example

10 5

2x x

Multiply by 2x because it is the least common denominator

10 52 2

2x xx x

20 5 25


Special cases-– Example

– Example

5 25 25 5x x x x All real

8 16 8x x 16 0 No solution

Properties

Addition Property of Equality If a=b then a+c = b+c and c+a = c+b

Properties

Multiplication Property of Equality If a,b,c are any real numbers and a=b then

ca=cb and ac=bc

Properties

Reflexive Property of Equality If a is a real number then a=a

Properties

Symetric property of equality a=b then b=a

Properties

Transitive property of equality If a=b and b= c then a=c

Properties

Associative property of Addition (a+b) + c = a + (b+c)

Properties

Associative property of multiplication (ab)c = a(bc)

Properties

Commutative Property of Addition a+b = b+a ab=ba

Properties

Commutative property of multiplication

4 5 5 4

Properties

Distributive Property a(b+c) = ab+ac

Properties

Prop. Of opposites or inverse property of addition

5+(-5)=0

Properties

Property of reciprocals or inverses prop. Of multiplication

For every nonzero real number a, there is a unique 1/a

1aa

1 and1

1aa

Properties

Identity property of addition There is a unique real number 0 such that for

every real number a a+0=a 0+a=0

Properties

Identity property of multiplication There is a unique real number 1 such that for

every real number a, 1a a and 1 a a

Properties

Multiplicative property of zero

0 0a 0 0a and

Properties

Closure property of addition For all real numbers a and b:

a+b is a unique real number

Properties

Closure property of Multiplication For all real numbers a and b:

ab is a unique real number

Properties

Product of powers property

5 4 9k k k

Properties

Power of a product property

7( )ab 7 7a b

Properties

Power of a power property

2 4 8( )a a

Properties

Quotient of powers property Subtract the exponents

53

2

xx

x

Properties

Power of a quotient property

3 3

3

a a

b b( )

Properties

Zero Power Property

0(4 ) 1ab

Properties

Negative power property

22

1a

a

Properties

Zero product property If (x+3)(x-2)=0, then (x+3)=0 or (x-2)=0

Properties

Product of roots property

20 4 5

Properties

Quotient of roots property

453

5

Properties

Root of a power property

33 xx

Properties

Power of a root property

2( 7) 49 7

Solving first power inequalities in one Variable

Examples of a first power inequalities-

– When something is equal to another number, then you use a dark circle, but when it isn’t equal to, you use a a non dark circle.

5x 5

2x 2


Disjunction– A Disjunction uses the

word or Example-

3 1x orx

1 3


Conjunctions– conjunctions include and

Example- x<3 and x>1 Or 3>x>1

1 3

Linear equations in two variables

Slope of lines– Horizontal: 0 – Vertical: Undefined– Linear: rise over run


Equations of lines– Slope intercept form- Y=mx+b – Standard form: ax+by=c– vertical X= a constant– Horizontal y=a constant


In order to graph a line you need– A point and slope– Or two point– Or an equation

2 1y x

slope Y intercept

Y intercept


How to find intercepts– X intercept- look for a point on the graph where y

equals zero– Y intercept- look for a point on the graph where x

equals zero


How and when to use the point slope formula-– You use the point slope formula when you don’t

know the y-intercept

y y

Linear systems

Substitution Method- – Example- 15

4 3 38

x y

x y

15x y 15y x

4 3(15 ) 38x x

Plug 15-x in for y

4 45 3 38x x 45 38x

7x

Linear systems

Addition and Subtraction Method (Elimination)

– Example-

5 12

3 4

x y

x y

Since the y’s already cross each other out there is no need to use the least common denominator

8 16 2x x

Linear systems

You can use graphing but it only gives an estimate

Linear systems

Check for understanding of terms-– Dependent system- Infinite set or all points (if

same line is used twice)– Inconsistent system-Null set (if they are parallel)– Consistent system-One point (if they cross)

Factoring

Methods– GCF- always look for the GCF first– Difference of Squares- used for binomials– Sum or Difference of cubes- used for binomials– PST- For trinomials– Reverse of FOIL- Trinomials – Grouping- Grouping

Factoring

GCF– Example

-

22 8 8x x 22( 4 4)x x

2( 2)( 2)x x

Factoring

Difference of Squares

4 275 108x y4 23(25 36 )x y

2 23(5x – 6y) (5x 6y)

Factoring

Sum or difference of cubes3 3x y

2 2( )( )x y x xy y

Factoring

Perfect Square Trinomial

2

2

4 4

( 2)

x x

x

Factoring

Reverse Foil-– Trial and error

2

2

2

ax bx c

ax bx c

ax bx c

( _ _ )( _ _ ) (_ _)(_ _) (_ _)(_ _)

Factoring

Grouping- – Example- 3 22 2b b ab a

2 ( 2) ( 2)b b a b

2( )( 2)b a b

Rational expressions

Simplify by factor and cancel-

2 ( 1)

1 1

x x x xx

x x

Rational Expressions

Addition and Subtraction of rational expressions– Addition-use LCM to cancel out the variable

2 1

4 5

6 6

1

a b

a b

b

b


Subtraction of rational expressions– Use LCM to cancel out the variables-– Example-

6 4 5

6 2 1

2 4

2

6 8 5

8 8

6 3

1

2

a b

a b

b

b

a

a

a


Multiplication and division of rational expressions– Example- 2 4

342

2

x y zxy

xyz

32xy

Quadratic equations in one variable

Solve by factoring– Example

2 2 8

( 2)( 4) 0

2 4

x x

x x

x x


Solve by taking the square root of each side– Example-

2

2

49 0

49 49

49

7

x

x

x


Solve by completing the square– Example-

2

2

2

2

6 2 0

6 ____ 2 _____

6 9 2 9

( 3) 11

3 11

x x

x

x x

x

x

Take half of x and square it


Quadratic formula– Example

2 4

2

b b ac

a

2 3 10 0

3 9 (40)

2

493

23 7

52

3 72

2

x x

or

Quadratic Equation


What does the discriminant tell you?– Discriminant is the value of

2 4b ac

2 4b ac

Functions

What does f(x) mean?– F(x)= name of independent variable or argument– Usually equal to “Y”– Not all relations are functions (those that are

undefined)– Ex.

2( ) 3 1f x x y

Functions

range and domain of a function Domain- set of all x values Range- set of all y values Ex. Ex.(2)

2

(0) 0

( ) 5 10

(0) (0,0)

f let x

f x x x

f

2

( ) 0 0

0 5 10

5 ( 2) 0

0 2

f x when y

x x

x x

x x

Functions

Ordered pairs– Ex. (1,1) (5,5)– Slope equals

5 11

5 1

1

1 1

0

y x b

b

b

y x

Functions

Quadratic functions How to graph a parabola

– If A>0 then it opens up– If A<0 then it opens down– Vertex- is equal to a –b/2a to find x – Plug into f(x) to find y– Axis of symmetry- vertical through the vertex so

x= -b/2a

Functions

How to graph a parabola cont. – Y int. let x=0 or f (0)– X int. let y=0 or f (x) (0)– Factor and find solutions

Simplifying expressions with exponents

A.) Product of powers

3 4 3 4 7.2 2 2 2

m n m na a a

ex


B.) quotient of powers

4 2 4 2 2. 2 2 2 2 4

m n m na a a

Ex


C.) Power of a Power

3 3 9

( )

. (2 ) 2 512

m n mna a

Ex


D.) Power of a Product

4 4 4 4

( )

. (2 ) 2 16

m m mab a b

Ex x x x


E.) Power of a Quotient

22

2

( )

4 4 16 1. ( )

8 8 64 4

mm

m

a a

b b

Ex

Simplifying expressions with radicals

A.) Root of a Power

B.) Power of a Root

3 3x x

2

2. 7 7

x x

Ex

Simplifying expressions with radicals

C.) Rationalizing the Denominator– Use the multiplication identity property

7 2 7 2. ( )

22 2Ex

Word Problems

Example 1-– A baseball game has 1200 people attending. Adult tickets are 5

dollars an student tickets are two dollars. The total amount of money made a tickets was 3660 dollars. The visiting team is entitled to half of the adult tickets sales. How much money does the visiting team get? 1200

1200

5 2 3660

5 2 2400 3660

3 1260

420

$1050

x y

y x

x y

x x

x

x adults

other school gets

Word Problems

Example 2-– Al left MUHS at 10:30 AM walking 4 mi/hr. Bob left MUHS

at noon running to catch up with Al. If Bob overtakes Al at 1:30 PM how fast was he running.

rate time distance

Al

Bob

4mi

hr

mibhr

3 hrs

3

2hrs

12 mi3

2bmi

Equal distance

Step 1- label variables Step 2- write an equation

312

2b

Step 3- solve for the variable2 2 3

123 3 2

8

b

b

Step 4

Bob’s rate- 8mi

hr

Word Problems

Example 3-– A serving of beef has 320 more calories than a serving of

chicken. The calories in 3 servings of beef is equal to the calories in seven servings of chicken. Find the number of calories in a serving of each meat.

:

: 320

chicken c

beef c 3( 320) 7c c

3 960 7

3 3

960 4

4 4240

c c

c c

c

c

: 240

: 560

chicken calories

beef calories

Word Problems

Example 4-– The length of a rectangle is 3 cm less then twice the width.

The perimeter is 34 cm more then the width. Find the length and width of the rectangle?

6 6 34

5 6 34

6 6

5 40

5 58

w w

w w

w

w

w

w w

2w-3

2w-3

6 6 34w w

6 6 34

5 6 34

6 6

5 40

5 58

w w

w w

w

w

w

8cm

13 cm

Line of Best fit or Regression line

You use to the line of best fit to estimate what the average is for the data

Your TI-84 calculator can determine the line of best fit for you

Line of Best fit or Regression line

What is the best fit line here?

Draw a line on the graph if you want.

Documents

Math Exam Review Semester 2 By Kyle Skarr and Ryan McLaughlin