Math 40
Chapter 6
Lecture Notes
Professor Miguel Ornelas
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M. Ornelas
Math 40 Lecture Notes
Section 6.1
Section 6.1
Simplifying Rational Expressions
Rational Expression
A rational expression is any expression that can be written in the form
P
, where P and Q are polynomials.
Q
Find all numbers for which the rational expressions are undefined.
a. f (x) =
x+5
x−7
b. g(x) =
x2
x+4
− 3x − 10
Simplifying Rational Expressions
Simplify each rational expression.
a.
8x2
24x
b.
5a + 15
10
c.
18y 2 + 6y
6y 2 + 15y
d.
3z 2 − 2z − 1
z 2 − 3z + 2
Section 6.1 continued on next page. . .
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c.
g(x) =
x2 + 3x − 4
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e.
x2 − 16
x2 − 8x + 16
Math 40 Lecture Notes
f.
Section 6.1 (continued)
x2 + xy + 2x + 2y
x+2
Section 6.2
Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions
Multiply.
a.
−7x2 3y 5
·
5y
14x2
b.
x2 + x
6
·
3x
5x + 5
c.
3x + 3 2x2 + x − 3
·
5x2 + 5x
4x2 − 9
d.
y 3 − 125
2y 2 + 17y + 21
·
2y 2 − 2y − 15 y 2 + 2y − 35
Section 6.2 continued on next page. . .
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Math 40 Lecture Notes
Section 6.2 (continued)
Dividing Rational Expressions
Divide.
1.
6y 3
42
÷
2
3y − 27 y − 3
2.
10x2 + 23x − 5
2x2 + 9x + 10
÷
5x2 − 51x + 10 7x2 − 68x − 20
Section 6.3
Adding and Subtracting Rational Expressions
Add or subtract.
a.
x 5x
+
7
7
b.
x2
16
−
x+4 x+4
c.
3 6
+
5 7
d.
♣ ♥
−
♦ ♠
Adding or Subtracting Rational Expressions with Unlike Denominators
1. Step 1
2. Step 2
3. Step 3
4. Step 4
5. Step 5
Section 6.3 continued on next page. . .
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Math 40 Lecture Notes
Add or subtract.
a.
3
7
+
10x2
25x
b.
c.
x+4 x−7
−
x−2 x+5
d.
2
5
+
3t t + 1
x2
2
x
− 2
+ 5x + 6 x + 3x + 2
Section 6.4
Solving Equations Containing Rational Expressions
1. Step 1
2. Step 2
3. Step 3
4. Step 4
5. Step 5
Section 6.4 continued on next page. . .
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Section 6.3 (continued)
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Math 40 Lecture Notes
Section 6.4 (continued)
Solve.
1.
5x 3
7x
− =
4
2
8
2.
6 x+9
2
−
=
x
5x
5
3.
6
y+3
=
y−3
y−3
4.
2x
6 − 2x
x
+
=
x − 3 x2 − 9
x+3
5.
1
3
1
+
=− 2
y 2 + 5y + 4 y 2 − 1
y + 3y − 4
6.
z
1
3
−
= 2
2z 2 + 3z − 2 2z
z + 2z
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Math 40 Lecture Notes
Section 6.5
Section 6.5
Rational Equations and Problem Solving
Finding an Unknown Number
1. The sum of a number and its reciprocal is
13
. Find the number.
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Distance, Rate, and Time Problems
1. The speed of a boat in still water is 20 miles per hour. It takes the same amount of time for the boat
to travel 3 miles downstream as it does to travel 2 miles upstream. Find the speed of the current.
Distance
Rate
Time
Upstream
Downstream
2. The speed of the wind is 3 miles per hour. It takes a cyclist a total of 3 hours to travel 12 miles against
the wind and return 12 miles with the wind. What is the speed of the cyclist with no wind?
Distance
Against Wind
With Wind
Section 6.5 continued on next page. . .
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Rate
Time
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Math 40 Lecture Notes
Section 6.5 (continued)
Work Problems
1. One hose can fill a goldfish pond in 45 minutes, and two hoses can fill the same pond in 20 minutes.
Find how long it takes the second hose alone to fill the pond.
hose 1 fills
Table 1: In 1 hour
hose 2 fills both hoses fill
2. Melissa can clean the house in 4 hours, whereas her husband Edward can do the same job in 5 hours.
They have agreed to clean together so that they can finish in time to watch a movie on TV that starts
in 2 hours. How long will it take them to clean the house together? Can they finish before the movie
starts?
Melissa completes
Table 2: In 1 hour
Edward completes Together they complete
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Math 40 Lecture Notes
Section 6.6
Simplifying Complex Fractions
Simplify each complex fraction.
a.
2x
27y 2
6x2
9
c.
b
1
+
a2
a
1
a
+
b2
b
b.
5x
x+2
10
x−2
d.
5
2
−
a a2
10
3
3−
+ 2
a
a
b.
4
a
=
a+2
2
3+
Section 6.7
Proportions
Solve each proportion.
a.
a
5
=
3
12
Section 6.7 continued on next page. . .
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Section 6.6
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Math 40 Lecture Notes
Section 6.7 (continued)
1. The Toyota Pruius is a hybrid car that travels approximately 204 mi on 4 gal of gas. Find the amount
of gas required for a 714-mi trip.
2. A sample of 220 compact fluorescent light bulbs contained 8 defective bulbs. How many defective bulbs
would you expect in a batch of 1430 bulbs?
3. If 100 grams of ice cream contain 13 grams of fat, how much fat is in 350 grams of ice cream?
Section 6.8
Variation and Problem Solving
Direct Variation
y varies directly as x , or y is directly proportional to x , if there is a nonzero constant k such that
y = kx
Section 6.8 continued on next page. . .
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M. Ornelas
Math 40 Lecture Notes
Section 6.8 (continued)
1. Suppose that y varies directly as x. If y is 17 when x is 34. Find y when x is 12.
2. Suppose that y varies directly as the square of x. When x is 4, y is 32. Find x when y is 50.
Inverse Variation
y varies inversely as x , or y is inversely proportional to x , if there is a nonzero constant k such that
y=
k
x
1. Suppose that y varies inversely as x. If y is 0.02 when x is 75. Find y when x is 30.
2. The intensity (I) of light from a source varies inversely as the square of the distance (d) from the
source. Ten feet away from the source the intensity is 200 candlepower. What is the intensity 5 feet
from the source?
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