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11.1 Problem Solving Using Ratios and Proportions
A ratio is the comparison of two numbers written as a fraction.
For example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses?
Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses.
The ratio is games won___________games lost
= 7 games_______3 games
= 7__3
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In a ratio, if the numerator and denominator are measured in different units then the ratio is called a rate.
A unit rate is a rate per one given unit, like 60 miles per 1 hour.
Example: You can travel 120 miles on 60 gallons of gas. What is your fuel efficiency in miles per gallon?
Rate = 120 miles________60 gallons= ________20 miles
1 gallon
Your fuel efficiency is 20 miles per gallon.
11.1 Problem Solving Using Ratios and Proportions
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An equation in which two ratios are equal is called a proportion.
A proportion can be written using colon notation like this
a:b::c:d
or as the more recognizable (and useable) equivalence of two fractions.
a___ ___=b
cd
11.1 Problem Solving Using Ratios and Proportions
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a:b::c:d a___ ___=b
cd
When Ratios are written in this order, a and d are the extremes, or outside values, of the proportion, and b and c are the means, or middle values, of the proportion.
Extremes Means
11.1 Problem Solving Using Ratios and Proportions
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To solve problems which require the use of a proportion we can use one of two properties.
The reciprocal property of proportions.
If two ratios are equal, then their reciprocals are equal.
The cross product property of proportions.
The product of the extremes equals the product of the means
11.1 Problem Solving Using Ratios and Proportions
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x
35
3
5
3535 x
1055 x
11.1 Problem Solving Using Ratios and Proportions
21x
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9
62
x
x 629
x618
x3
11.1 Problem Solving Using Ratios and Proportions
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Solve:1
21
xx
x – 1 = 2x
x = –1
xx 2)1(1
11.1 Problem Solving Using Ratios and Proportions
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Solve:xx
x 1
12
x2 = -2x - 1
x2 +2x + 1= 0
)12(12 xx
11.1 Problem Solving Using Ratios and Proportions
(x + 1)(x + 1)= 0
(x + 1) = 0 or (x + 1)= 0 x = -1
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11.2 Problem Solving Using Percents
Percent means per hundred, or parts of 100 When solving percent problems, convert the
percents to decimals before performing the arithmetic operations
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11.2 Problem Solving Using Percents
• What is 20% of 50?• x = .20 * 50• x = 10
• 30 is what percent of 80?• 30 = x * 50• x = 30/50 = .6 = 60%
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11.2 Problem Solving Using Percents
• 12 is 60% of what?• 12 = .6x• x = 12/.6 = 20
• 40 is what percent of 300?• 40 = x * 300• x = 40/300 = .133… = 13.33%
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11.2 Problem Solving Using Percents
What percent of the region is shaded?
60
40
10
10
100 is what percent of 2400?
100 = x * 2400?
x = 100/2400
x = 4.17%
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11.3 Direct and Inverse Variation
Direct VariationThe following statements are equivalent:
y varies directly as x. y is directly proportional to x. y = kx for some nonzero constant k.
k is the constant of variation or the constant of proportionality
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11.3 Direct and Inverse Variation
Inverse Variation
The following statements are equivalent:
y varies inversely as x. y is inversely proportional to x. y = k/x for some nonzero constant k.
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11.3 Direct and Inverse Variation
If y varies directly as x, then y = kx.If y = 10 when x = 2 , then what is the value of y when x = 8?x and y go together. Therefore, by substitution 10 = k(2).What is the value of k? 10 = 2k
10 = 2k
5 = k
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11.3 Direct and Inverse Variation
k = 5
Replacing k with 5 gives us y = 5x
What is y when x = 8 ?
y = 5(8)
y = 40
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11.3 Direct and Inverse Variation
If y varies inversely as x, then xy = k.
If y = 6 when x = 4 , then what is the value of y when x = 8?
x and y go together. Therefore, by substitution (6)(4) = k.
What is the value of k?
24 = k
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11.3 Direct and Inverse Variation
k = 24
Replacing k with 24 gives us xy = 24
What is y when x = 8 ?
8y = 24
y = 3
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y = kx
00 5 10 15 20
5
10
15
Direct variation
11.3 Direct and Inverse Variation
y = 2x
••
••
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xy= k
00 5 10 15 20
5
10
15 •
••
• •
xy= 16
Inverse Variation
11.3 Direct and Inverse Variation
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11.5 Simplifying Rational Expressions
Define a rational expression. Determine the domain of a rational
function. Simplify rational expressions.
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Rational numbers are numbers that can be written as fractions.
Rational expressions are algebraic fractions of the form P(x) , where P(x) and Q(x) Q(x) are polynomials and Q(x) does not equal zero.
Example:
3x 2 2x 1
4 x 1
11.5 Simplifying Rational Expressions
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P(x) ; Since division by zero is not Q(x) possible, Q(x) cannot equal zero.
The domain of a function is all possible values of x.
For the example , 4x + 1 ≠ 0
so x ≠ -1/4.
14
123 2
x
xx
11.5 Simplifying Rational Expressions
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The domain of is
all real numbers except -1/4.
14
123 2
x
xx
Domain = {x|x ≠ -1/4}
11.5 Simplifying Rational Expressions
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Find domain of 65
122
xx
x
Domain = {x | x ≠ -1, 6}
Solve: x 2 –5x – 6 =0
(x – 6)(x + 1) = 0The excluded values are x = 6, -1
11.5 Simplifying Rational Expressions
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To simplify rational expressions, factor the numerator and denominator completely. Then reduce.
Simplify:
32244
12222
2
xx
xx
11.5 Simplifying Rational Expressions
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864
62
32244
12222
2
2
2
xx
xx
xx
xxFactor:
244
232
xx
xxReduce:
2
42
3
x
x
11.5 Simplifying Rational Expressions
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Simplify:
x
x
2
2
Factor –1 out of the denominator: x
x
21
2
21
2
x
x
11.5 Simplifying Rational Expressions
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Reduce: 21
2
x
x
11
1
11.5 Simplifying Rational Expressions
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11.5 Simplifying Rational Expressions
Multiply rational expressions. Divide rational expressions
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To multiply, factor each numerator and denominator completely.
Reduce Multiply the numerators and multiply the
denominators. Multiply:
213
12
209
15 2
2
2
x
xx
xx
x
11.5 Simplifying Rational Expressions
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xx
xx
xx
x
213
12
209
152
2
2
2
Factor:
73
34
54
15 2
xx
xx
xx
xReduce: 5
7
3
5
5
x
x
x
x
11.5 Simplifying Rational Expressions
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7
3
5
5
x
x
x
xMultiply: 75
35
xx
xx
3512
1552
2
xx
xx
11.5 Multiplying and Dividing
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To divide, change the problem to multiplication by writing the reciprocal of the divisor. (Change to multiplication and flip the second
fraction.)
Divide:
54
62
1
322
2
2
2
xx
xx
x
xx
11.6 Multiplying and Dividing
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62
54
1
322
2
2
2
xx
xx
x
xx
54
62
1
322
2
2
2
xx
xx
x
xx
Change to multiplication:
Factor completely:
232
51
11
132
xx
xx
xx
xx
11.5 Multiplying and Dividing
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232
51
11
132
xx
xx
xx
xxReduce:
2
5
x
xMultiply:
11.5 Multiplying and Dividing
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11.7 Dividing Polynomials
Dividing a Polynomial by a MonomialLet u, v, and w be real numbers, variables or
algebraic expressions such that w ≠ 0.
w
v
w
u
w
vu
.1
w
v
w
u
w
vu
.2
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11.7 Dividing Polynomials
x
xxx
3
9612 23 324 2 xx
c
ccc
9
452718 24 532 3 cc
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11.7 Dividing Polynomials
)2()124( 2 xxx Use Long Division
1242 2 xxx
x
x2 -2x6x - 12
+ 6
6x - 12
0
Note: (x + 6) (x – 2) =x2 + 4x - 12
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11.7 Dividing Polynomials
)1()24( 2 xxx Use Long Division
141 2 xxx
x
x2 - x5x - 1
+ 5
5x - 5 4
1
4545
xxx orr
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11.7 Dividing Polynomials
)2()12( 3 xxx Note: x2 term is missing
1202 23 xxxx
x2
x3 + 2x2
-2x2 + 2x
- 2x
-2x2 – 4x 6x - 1
-
+ 6
6x + 12-
-13
13622 rxx
2
13622
x
xxor
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xx
12
2
13
LCD: 2x
Multiply each fraction through by the LCD
x
xx
x
x 12*2
2
23*2
246 x
18 x
18x Check your solution!Check your solution!
18
12
2
1
18
3
1293
11.8 Solving Rational Equations
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Solve. 1
54
1
5
xx
xLCD: ?LCD: (x+1)
)1(
)1(5)1(4
)1(
)1(5
x
xx
x
xx
5445 xx145 xx
1x
Check your solution!Check your solution!
11
54
11
)1(5
0
54
0
5
?
No Solution!No Solution!
11.8 Solving Rational Equations
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Solve. 14
6
2
232
xx
x
Factor 1st!
1)2)(2(
6
2
23
xxx
x
LCD: (x + 2)(x - 2)
)2)(2()2)(2(
)2)(2(6
)2(
)2)(2)(23(
xx
xx
xx
x
xxx
42264263 22 xxxxxx2443 22 xxx
0642 2 xx0322 xx
0)1)(3( xx01or 03 xx
1or 3 xx
Check your solutions!Check your solutions!
11.8 Solving Rational Equations
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Short Cut!
When there is only fraction on each side When there is only fraction on each side of the =, just cross multiply as if you are of the =, just cross multiply as if you are solving a proportion.solving a proportion.
11.8 Solving Rational Equations
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Example: Solve.
4
1
4
32
xxx
xx 42 12342 xxx
0122 xx
0)3)(4( xx
03or 04 xx
3or 4 xx
Check your solutions!Check your solutions!
)4(3 x
11.8 Solving Rational Equations
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Solve.1
2
22
62
x
x
xx
)1(6 x)2)(1(2 xxx
6)2(2 xx3)2( xx
0322 xx
0)1)(3( xx
01or 03 xx
1or 3 xx
1
2
)1(2
6
x
x
xx
Check your solutions!Check your solutions!
11.8 Solving Rational Equations