Let c be any real number. If a = b, then a + c = b + cIf a = b, then a – c = b – c If a = b, then a ( c )= b ( c )If a = b, then a /c = b /c x + 7 – 7.

Let c be any real number.

If a = b, then a + c = b + c If a = b, then a – c = b – c

If a = b, then a ( c )= b ( c ) If a = b, then a /c = b /c

x + 7 – 7 = 22 – 7 x = 15

– 7 = – 7

x = 15Vertical steps = less writing.

+ 12 = + 12

2 2

5 5

41x 9x 26x

Direction: “SOLVE.” this means, WORK BACKWARDS

“UNDO” the operations on the x term..

Rules to follow:SIMPLIFY to 1 variable term. You may need to combine like terms, distributive property to remove ( )’s, and Clear Fractions by multiplying both sides by the LCD (least common denominator).

1753 x 173

4x

1345 x 18.82.73.16 x

– 5 = – 5

-7.2 -7.2

4.3x

123 x

– 45 = – 45

3 3 4x 4 4

3 3

+7 = +7

83

4x

2

6x

– 16.3 = – 16.3

32 x-1 -1

32x

48.242.7 x

Sect 2.2Solving using Order of Operations

1443 xx

Combining Like Terms in the simplifying stage.

685 xx 7410756 xxx

7 7

+ 8x = + 8x 147 x 657 x

– 5 = – 5 17 x

7 7

7

1x

xx 41715 + 4x = + 4x

1735 x – 5 = – 5

123 x

3 3

2x

4x

Always move the smallest variable term!

123552 xx

Distributive Property in the simplifying stage.

1632552 xx

73235 xx + 5x =+ 5x

7823 x+ 7 = + 7

x816 8 8

x 2

5

To clear fractions, multiply both sides by the LCD.

LCD = 6

8235

2xxx 2

6

1

3

26 6

6 2

xx 1214 – 4x = – 4x

x818 8

x8

1

Two approaches.

85

4

5

6x

4046 x

366 x

6x

8235

2x

2023 x

183 x

5 5

– 4 = – 4

6 6

4 52

52

– 2 = – 2

3 3

6x

To clear fractions, multiply both sides by the LCD.

3

2

4

3

2

3

2

1x

LCD = 12

12 12 12

12 6

34

6

89186 x

896 x

16 x6

1x

– 9 = – 9

6 6

To clear decimals, multiply both sides by 10. Every multiplication of 10 moves the decimal one place to the right. We will multiply by 100.

10094.12.13.5100 x

194120530 x

– 530 = – 530

336120 x120 = 120 8.2x

All the above examples are Conditional equations…they have a solution. Contradiction Equation. (No solution) Identity Equations. (Have infinite sols.)

xxx 2253 xxx 51772

xxx 4253

4353 xx

45

xxx 57772

7272 xx

77 False Statement

No SolutionContradiction Equation

True StatementInfinite SolutionsIdentity Equation

bhA2

1 WLP 22 Solve for b.

22

12 bhA

bhA2h h

bh

A

2

Isolate the b by undoing the operations taking place on the variable b.

Solve for W.

Remove the terms with no W 1st. Next isolate the W by undoing the operations taking place on the variable W.

WLP 22

WLP

2

2

– 2L = – 2L

2 2

222 rrhA Solve for a.

22 rMove the terms and factors that are outside the ( ) ‘s to the other side.

Solve for h.

Remove the terms with no h 1st. Next isolate the h by undoing the operations taking place on the variable h.

ahwK 6917

ahwK 6917

– 917 = – 917

6 6

ahwK

6

917

Since we have – a, add the a to the left and subtract the fraction to the right. a will be isolated.

aKK

a

6

917

6

917

6

917K

hwa

22 r

rhrA 22 2 r2r2

hr

rA

2

2 2

Sect 2.4Solving Percentage Problems

01.0100

1

100nn

n

n% =

Convert decimals (fractions) to percent and percent to decimal.

100

%

d

nFraction to a percent Convert to a percent.

5

3

100

x

Cross multiply and products are equal.

x5 300

5

300

5

5x

60x

%60

Decimal to a percent. Multiply by 100

Convert to a percent.

359.0 100 %9.35

Notice we moved the decimal point 2 places to the right.

Percent to a decimal. Drop % and divide

by 100

%9.19

Convert to a decimal.

199.01009.19

Notice we moved the decimal point 2 places to the left.

We will work both techniques…pick your favorite.

x 01.011 39.5x

49

of

is

100

x

4911

Not to bad… just like Sect 1.1 and 1.2.

What is 11% of 49?

The values that associates with the words “is” and “of”

4911100 x100 100

39.5x

Notice we will always cross multiply by the numbers on the diagonal and divide by the 3rd number!

3 01.16

16.0

x

x75.18163100 x

16.0 of

is

100 x316

Cross multiply the numbers on the diagonal and divide by the 3rd number! Calculator !!!

75.18

34 50

50.0

x

68x50.0

Here is where problems may occur! x really needs to multiplied by 0.01 because of the word percent!

3450.0 x

%68

5034100 x

of

is

100

x 34

50Cross multiply the numbers on the diagonal and divide by the 3rd number! Calculator !!!

68

%68

x*16.03

01.

In 2006, there were 300 million people in the United States, and 62.2% of them lived within 5 mi. of a Wal-Mart store. How many people lived with 5mi. of a Wal-Mart store? Any number that represents a TOTAL associates with 100% and is

across from 100 in the proportion or associates with “of.”

of

is

100million1003002.62 x

x300

2.626.186

About 1.6 million students who graduated high school went to college. This was 66% of all high school graduates. How many total high school graduates are there? TOTAL is across from 100 in the

proportion or associates with “of.”

of

is

100million666.1100 x

6.1

x66

42.2

A car dealer lowered the sticker price of a car from $20,830 to $18,955.What percent of the regular price does the sale price represent?What is the percent discount?

The total bill was $47.70 that included 6% sales tax. How much was the merchandise before tax?

Any number that represents a ORIGINAL PRICE associates with 100% and is across from 100 in the proportion or associates with “of.”

of

is

1002083018955100 x

18955

20830

x563784.90

%6.90What is the percent discount? %4.9%6.90%100

of

is

100

6 70.47

x

We have TOTAL and ORIGINAL price…which is the “of”?

ORIGINAL price is the “of”? The $47.70 represent the ORIGINAL price + 6% sales tax

106 WAIT a minute! That means our PERCENTAGE is not 6%, 106%!

10670.47100 x 45 45$

Sect 2.5 Word Problems

Sect 2.5 Word Problems

Sect 2.5 Word Problems

78210 x8010 x 8x

70235 x701015 x

5510 x

5.5x

34342 x3468 x

266 x3

13

6

26x

22

1010

1515

1010

88 66

Sect 2.5 Word Problems

P S F

km260

x

Unknown distance = x3 times the unknown distance = 3x

3x

2603 xx2604 x

65x

How far he biked = 3x

How far he had left to go = x

kmx 65

km195653

Sect 2.5 Word Problems

1 mile

Milex

Milex + 1

First marker = x

Next consecutive marker = x + 1

x + x + 1 = 559

55912 x

5582 x279x

279x

2801279

Sect 2.5 Word Problems

Length

Length

WidthWidth

WLP 22

WL 22288

44 WL

44 WL

WL 22288

WW 2442288

WW 2882288

884288 W

W4200

W50

ft50

ft944450

94 ft by 50 ft

W W

Sect 2.5 Word Problems

B = Number of brochures to print

Starting Cost + The number of Brochures 3000$

2($300) + B(the cost of each Brochures) 3000$

$600 + B($0.215) 3000$Change cents to dollars!

2400215.0 B

8.11162B

Can print 11,162 brochures and not go over budget.

Sect 2.5 Word Problems

Back side is the unknown = x

Peak = 2x

Front = x + 20

The sum of three angles of a triangle is 180 degrees.

x + 2x + x + 20 = 180

4x + 20 = 180

4x = 160x = 40

40 80402

602040

180608040

Sect 2.5 Word Problems

H = the hammer price of final bid.

HH 108.01150 $1150 + 8%(of final bid) = final bid

H92.01150

H1250

The final bid must be $1250 or higher.

Proportion Style.Jared has to pay the Auctioneer 8% of the final bid, so he gets 92% of the final bid.

of

is

100

1150

H92

1250921150100 H

Let A > B and C is a non-zero positive constant.

BA CC CBCA

BA CC CBCA

BA CC CBCA

BA CC CBCA

C

B

C

A

REMEMBER!!!

How to write a solution set with set builder notation, interval notation and Graph.

Let x > 3 be our solution.

We want all x values between -2 and 3.

Graph.

0 3 3 (

Author. Mr. Fitz

Let x < 3 be our solution. 3| xx

0 3 3]

Author.Mr. Fitz

Compound inequalitiesLet x > – 2 and x < 3 be our solution.

x 2 3x0 3

]

Author.2

(3

Mr. Fitz2

“The set of all x, such that x > 3.”

32| xx

{ x | x > 3 }

Set builder notation

REMEMBER!When x is on the left side the inequality symbol points in the direction of the graph.

Interval notation 3,

Interval notation

,3

Interval notation 2,3

5213 xx 74

1x 182 x

Solve the inequalities. Graph the solution and write in interval notation.

51 x

4x

, 4

4

28x

, 28

28

9x

9,

9

Flip inequality

symbol

4

+ 1

– 2x– 2x

+ 1

4

– 2 – 2

756 x 1792 xx 652193 xx– 6– 6

15 x

5

1x

– 5 – 5

5

1

15,

– 2x– 2x

159 x

x510

x 2

2x

2, 2

If you move the smallest variable term, it will stay positive.

Switch the inequality around so x is on the left side.

– 1– 1

5 5

30521273 xx285283 xx

28288 x

08 x

0x

+ 5x+ 5x

+ 28 + 28

8 8

,0 0

$50 + $15(per person) < $450

4501550 p

Let p = how many people

40015 p 6.26pThe party cannot exceed 26 people!– 50– 50

15 15

Let h = how many hours worked in the 4th week

Average of 4 wks > 16 hours/week

164

141220

h44

6446 h– 46– 46

18hDina has to work at least 18 hours in the 4th week.