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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.