What is the Lowest Common Denominator (LCD)? 5.3 – Addition & Subtraction of Rational Expressions.
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Transcript of What is the Lowest Common Denominator (LCD)? 5.3 – Addition & Subtraction of Rational Expressions.
6
5
3
2and
What is the Lowest Common Denominator (LCD)?
2
1
7
2,
4
3and
3 5
5 13and
x x4 2
17 3
4 6and
y y
6 28
5x 12 4y
5.3 – Addition & Subtraction of Rational Expressions
5
7
5
3
a
aand
a
a
What is the Lowest Common Denominator (LCD)?
123
5
4
72
2
x
xand
x
x
5a
3
27xand 5x
4x 4x
243 x
4x 4x 3 4x 5a
5.3 – Addition & Subtraction of Rational Expressions
23
3
32
522
yy
yand
yy
y
What is the Lowest Common Denominator (LCD)?
1y
5y
3y 2y
3y 1y and
3y
1y 2y
5.3 – Addition & Subtraction of Rational Expressions
73
7
73
3
xx
x
Examples (Like Denominators):
1
3 7x 3x 7
5.3 – Addition & Subtraction of Rational Expressions
2
64
2
52 2
x
x
x
xx
Examples (Like Denominators):
2x 22 5
2
x x
x
22
2
x
x
2x
32 x22 5x x 4 6x
4x 6
x 6
2x 2 3x
5.3 – Addition & Subtraction of Rational Expressions
65
72
65
1322
xx
x
xx
x
Examples (Like Denominators):
65
72132
xx
xx
65
132
xx
x x2 7
65
62
xx
x
6x
x 1 x 6
1
1
x
1
1
5.3 – Addition & Subtraction of Rational Expressions
Examples:
LCD 15
5.3 – Addition & Subtraction of Rational Expressions
𝑦5−
4 𝑦15
( 33 ) 𝑦5 − 4 𝑦
15
3 𝑦15−
4 𝑦15
3 𝑦−4 𝑦15
−𝑦15
Examples:
210
11
8
5
xx
LCD 40x2
210
11
8
5
xx
x
x
5
5
22 40
44
40
25
xx
x 240
4425x
x
4
4
5.3 – Addition & Subtraction of Rational Expressions
Examples:
1
2
7
5
xx
LCD
5 2
7 1x x
7 1 7 1x x x x
7 1x x
7x 1x
1
1
x
x
7
7
x
x
5x 5 14x 19x 5
5.3 – Addition & Subtraction of Rational Expressions
3
5
9
102
xx
xExamples:
10x
10 5
3 3 3
x
x x x
10
3 3
x
x x
3 3x x
3 3x x
3 3x x
3
5
x
3x 3x
LCD
3
3
x
x
3 3x x 5x 15
10x 155x
5x 15
5 3x 3x 3x
5
3x
5.3 – Addition & Subtraction of Rational Expressions
812
3
23
42
x
x
xx
Examples:
LCD
4
4 3
3 2 4 3 2
x
x x x
x 3 2x 3x
4 3 2x
4 x 3 2x
4
4
x
x
5.3 – Addition & Subtraction of Rational Expressions
Examples: continued
4 3 2 4 3 2x x x x
4 3 2x x
x
x
x
x
xx 234
3
23
4
4
4
16 23x
216 3x
5.3 – Addition & Subtraction of Rational Expressions
444
622
x
x
xx
xExamples:
6x x
LCD
6
2 2 2 2
x x
x x x x
2x 2x 2x 2x
2x 2x 2x
2
2
x
x
2
2
x
x
5.3 – Addition & Subtraction of Rational Expressions
Examples: continued
2
2
2222
6
2
2
x
x
xx
x
xx
x
x
x
2 2 2 2 2 2x x x x x x
222
2126 22
xxx
xxxx
2 2 2x x x 22 2x x
26x 12x2x 2x
27x 10x x 7x 10
5.3 – Addition & Subtraction of Rational Expressions
Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions.
7
5
yx
xyyx
873
4 3
1
1
xx
x
5.4 – Complex FractionsComplex Fractions
3 24 31 32 8
3 24 31 32 8
3 43 4
4
3 24 3
1 384 2
24 24
2
3 24 31
43
2 824
9 812 12
4 38 8
6 3 8 2
12 1 3 3
11278
18 16
12 9
2
212
21
LCD: 12, 8 LCD: 24 24
8
7
12
1
7
8
12
12
3
24
5.4 – Complex Fractions
LCD: 6xy
56
3
yy xyx
2
2 2
5 6
2 6
x y
xy x y
56
3
6 6
6 6
yy x
xy xy
yy
xx xy
25 6x y2xy 3y x
xy
xy
y
3
65
6xy6xy
5.4 – Complex Fractions
LCD:
3759
3759
3 5
7 9
6337
6359
3 9
7 5
9 3
7 5
27
35
27
35
63
Outers over Inners
3759
27
35
)5)(7(
)9)(3(
5.4 – Complex Fractions
5.5 – Equations with Rational Expressions
5 16 1x 4 1
4 5 20
x
4 1
4 520 20
2020
x
5x
LCD: 20
5 16 1x
5 15x
3x
44 11
LCD:
2 3 3 3 2x x 2
2 3 2
3 3 9x x x
2 3 2
3 3 3 3x x x x
3 3x x
3 32
3xx x
2 3 3 3 2x x
2 6 3 9 2x x
5 3 2x
5 5x 1x
33
33
xx x
33 3
32
x xx x
5.5 – Equations with Rational Expressions
LCD: 6x5 3 3
3 2 2x
65
3x
2 5 3 3 3 3x x 10 9 9x x
9x
36
2xx
26
3x
10 9 9x x
5.5 – Equations with Rational Expressions
LCD: x+36 22
3 3
xxx x
3x x
2 3x x
2 12 0x x 3x
3 0 4 0x x 3x
36
3xx
2
33
xx
x
3 2x
6 2x 2 6x 2 3x x 6 4 6x
0 4x 4x
5.5 – Equations with Rational Expressions
LCD:
2
5 11 1 12
2 7 10 5
x
x x x x
5 11 1 12
2 2 5 5
x
x x x x
2 5x x
2 55
2xx x
5 5 11 1 12 2x x x
5 25 11 1 12 24x x x
5 25 23x x
6 48x 8x
1
2 52
1 15
x
xx
xx
2 5
12
5x
xx
5.5 – Equations with Rational Expressions
LCD: abx1 1 1
a b x
1abx
a
bx
bx ab ax
bx
bx
b x
Solve for a
1
babx 1
xabx
ax ab
a b x
a
5.5 – Equations with Rational Expressions
Problems about NumbersIf one more than three times a number is divided by the number, the result is four thirds. Find the number.
3x
33 1
xx
x
3 3 1 4x x
9 3 4x x
5 3x 3
5x
LCD = 3x1x
4
3
3
34
x
9 4 3x x
5.6 – Applications
Problems about Work
Mike and Ryan work at a recycling plant. Ryan can sort a batch of recyclables in 2 hours and Mike can sort a batch in 3 hours. If they work together, how fast can they sort one batch?
Time to sort one batch (hours)
Fraction of the job completed in one hour
Ryan
Mike
Together
1
21
31
x
2
3
x
5.6 – Applications
Problems about Work Time to sort one
batch (hours)Fraction of the job completed in one hour
Ryan
Mike
Together
1
2
1
31
x
2
3
x
1
2 6
1
2x
3x 5 6x 6
5x hrs.
LCD =1
3
1
x 6x
36
1x 1
6x
x
2x 61
15
5.6 – Applications
James and Andy mow lawns. It takes James 2 hours to mow an acre while it takes Andy 8 hours. How long will it take them to mow one acre if they work together?
Time to mow one acre (hours)
Fraction of the job completed in one hour
James
Andy
Together
2
8
x
1
21
81
x
5.6 – Applications
Time to mow one acre (hours)
Fraction of the job completed in one hour
James
Andy
Together
2
8
x
1
2
1
81
x
LCD:1
2 8
1
2x
4x 5 8x 8
5x
hrs.
1
8
1
x 8x
88
1x 1
8x
x
x 83
15
5.6 – Applications
A sump pump can pump water out of a basement in twelve hours. If a second pump is added, the job would only take six and two-thirds hours. How long would it take the second pump to do the job alone?
1
121
x
26
3
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together
x
12
20
3
5.6 – Applications
1203
1
12
1
x2
63
1203
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together
x
12
1
121 1
12 x
20
3
1
x 1
203
3
20
5.6 – Applications
Distance, Rate and Time Problems If you drive at a constant speed of 65 miles per hour and you travel for 2 hours, how far did you drive?
d r t
65miles
hour 2 hours 130 miles
dt
r
dr
t
5.6 – Applications
A car travels six hundred miles in the same time a motorcycle travels four hundred and fifty miles. If the car’s speed is fifteen miles per hour faster than the motorcycle’s, find the speed of both vehicles.
Rate Time Distance
Motor-cycle
Car
x
x + 15
450 mi
600 mi
t
t
r
dt
x
450
15
600
x
5.6 – Applications
Rate Time Distance
Motor-cycle
Car
x
x + 15
450 mi
600 mi
t
t
r
dt
x
450
15
600
x
x
450 15
600
xLCD: x(x + 15)
15
600450
xx
x(x + 15) x(x + 15)
5.6 – Applications
15
600450
xx
x(x + 15) x(x + 15)
xx 60045015
xx 60015450450
x15015450
x
150
15450
x45
45x mph
Motorcycle
6015 x mph
Car
5.6 – Applications
Rate Time Distance
UpStream
DownStream
A boat can travel twenty-two miles upstream in the same amount of time it can travel forty-two miles downstream. The speed of the current is five miles per hour. What is the speed of the boat in still water? boat speed = x
x - 5
x + 5
22 mi
42 mi
t
t
r
dt
5
22
x
5
42
x
5.6 – Applications
Rate Time Distance
UpStream
DownStream
boat speed = x
x - 5
x + 5
22 mi
42 mi
t
t
r
dt
5
22
x
5
42
x
5
22
x
5
42
xLCD: (x – 5)(x + 5)
5
42
5
22
xx(x – 5)(x + 5) (x – 5)(x + 5)
5.6 – Applications
542225 xx
2104211022 xx
xx 2242210110
x20
320
x16 16mph
Boat Speed
5
42
5
22
xx(x – 5)(x + 5) (x – 5)(x + 5)
x20320
5.6 – Applications
Dividing by a Monomial
where 0a b a b
cc c c
3 2
2
25 5
5
x x
x
3
2
25
5
x
x5x
8 6 4
3
21 9 12
3
x x x
x
8
3
21
3
x
x57x
2
2
5
5
x
x 1
6
3
9
3
x
x
4
3
12
3
x
x
33x 4x
5.7 – Division of Polynomials
Dividing by a Monomial
7 2 224 12 4 4x x x x
3 312 18 6
3
x y xy y
xy
7 2
2
24 12 4
4
x x x
x
7
2
24
4
x
x56x
3 312
3
x y
xy
2 24x y
2
2
12
4
x
x 2
4
4
x
x 3
1
x
18
3
xy
xy
6
3
y
xy
62
x
5.7 – Division of Polynomials
285
5
Review of Long Division
783 4
4 7835 2855
253 5
7
350
43 836
2
1 9
3
5
203
5.7 – Division of Polynomials
4
3195
4
3195
35125 2 xxx
x
xx 52 x7 35
7
357 x 0
5x 7x 35122 xx
Long Division
5
35122
x
xx
5.7 – Division of Polynomials
15023 2 xxx
x2
xx 62 2 x6 15
6
186 x 3
3x
3
362x
x 152 2 x
3
3
x
Long Division
3
152 2
x
x
5.7 – Division of Polynomials
72812 2 xxx
x4
xx 48 2 x6 7
3
36 x 4
12 x
12
434
xx 728 2 xx
12
4
x
Long Division
12
728 2
x
xx
5.7 – Division of Polynomials