Transcript Slide 1
Homework, Page 253
Solve the equation algebraically. Support your answer numerically
and identify any extraneous solutions.
1.
x2 x5 1
3
3
3
x2 x5 1
x 2 x 5 1
3
3
3
2 x 3 1 2 x 2 x 1
1 2 1 5 1
3 4 1
3 4 1
3
3
3
3 3 3
3
3
x 1
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Slide 2- 1
Homework, Page 253
Solve the equation algebraically. Support your answer numerically
and identify any extraneous solutions.
4x
12
5. x x 3 x 3
4x
12
4x
12
x
x 3 x
x 3 x 3
x
3
x
3
x 2 3x 4 x 12 x 2 x 12 0 x 3 x 4 0
4 4
12
16
12
x 4,3 4
4
4 3 4 3
7
7
4 4
28 16
12
12
3
x 4
7 7
7
3 3 4 3
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Slide 2- 2
Homework, Page 253
Solve the equation algebraically and graphically. Check for
extraneous solutions.
12
9. x 7
x
12
12
x 7 x x 7 x 2 12 7 x x 2 7 x 12 0
x
x
12
12
x 3 x 4 0 x 3, 4 3 7 4 7
3
4
x 3, 4
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Slide 2- 3
Homework, Page 253
Solve the equation algebraically. Check for extraneous solutions.
Support your answer graphically.
13.
3x
1
7
2
x 5 x 2 x 3 x 10
1
7
3x
2
x 3x 10 3 x x 2 x 5 7
2
x 5 x 2 x 3 x 10
3 x 2 6 x x 5 7 3 x 2 5 x 2 0 3 x 1 x 2 0
1
1
7
1
x , 2
2
3 1 5 1 2 1
1
3 10
3
3
3
3
3 3
63
21 42
63
1
x
14 7
98
98 98
98
3
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Slide 2- 4
Homework, Page 253
Solve the equation algebraically. Check for extraneous solutions.
Support your answer graphically.
3
6
3 x
2
17.
x 2 x 2x
x
6
3 x
3
2
x x 2
x
x 2 x 2x
3 x 6 3 x x 2 3 x 6 x 2 x 6
x 2 2 x 0 x x 2 0 x 0, 2
No solution
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Slide 2- 5
Homework, Page 253
Two possible solutions to the equation f (x) = 0 are listed. Use
the given graph of y = f (x) to decide which, if any are extraneous.
21. x 2 or x 2
Both are extraneous.
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Slide 2- 6
Homework, Page 253
Solve the equation.
25.
x2 2 x 1
0
x5
x2 2 x 1
2
2
0 x 2 x 1 0 x 1 0
x5
x 1
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Slide 2- 7
Homework, Page 253
Solve the equation.
29.
5
x 8
x
2
x 3.100,0.661, 2.439
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Slide 2- 8
Homework, Page 253
33. Mid Town Sports Apparel has found that it needs to
sell golf hats at $2.75 each to be competitive. It costs
$2.12 to produce each hat, and weekly overhead is
$3,000.
a. Let x be the number of hats produced each week.
Express the average cost (including overhead) of
producing one hat as a function of x.
b. Solve algebraically to find the number of golf hats
that must be sold each week to make a profit.
Support your answer graphically.
c. How many golf hats must be sold to make a profit of
$1,000 in one week. Explain your answer.
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Slide 2- 9
Homework, Page 253
33. a. Let x be the number of hats produced each week.
Express the average cost (including overhead) of
producing one hat as a function of x.
2.12 x 3000
c x
x
b. Solve algebraically to find the number of golf hats
that must be sold each week to make a profit.
2.12 x 3000
2.12 x 3000
c x
2.75
x
x
2.75 x 2.12 x 3000 0.63x 3000 x 4762
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Slide 2- 10
Homework, Page 253
33. c. How many golf hats must be sold to make
a profit of $1,000 in one week. Explain your
answer.
1000
x 4762
x 6,350
2.75 2.12
To break even, 4,762 hats must be sold. After
breaking even, a profit of $0.63 is made on each
hat sold, so selling a total of 6,350 hats will yield
a $1,000 profit.
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Slide 2- 11
Homework, Page 253
37. Drake Cannery will pack peaches in 0.5-L cylindrical
cans. Let x be the radius of the can in cm.
a. Express the surface area of the can S as a function of x.
500
2
V 500 r h h 2 S 2 r 2 rh
r
2
500
1000
2
S x 2 x 2 x 2 S x 2 x
x
x
2
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Slide 2- 12
Homework, Page 253
37. b. Find the radius and height of the can if the surface
area is 900 cm2.
1000
900 2 x
x 1.121,11.368
x
500
500
h
253.327 h
2.463
2
2
1.121
11.368
2
r 1.121cm; h 253.327cm or r 11.368cm; h 2.463cm
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Slide 2- 13
Homework, Page 253
41. Drains A and B are used to empty a swimming
pool. Drain A alone can empty the pool in 4.75 h. Let t
be the time it takes for drain B alone to empty the pool.
a. Express as a function of t the part D of the drainage
that can be done in 1 h with both drains open at the
same time.
b. Find graphically the time it takes for drain B alone to
empty the pool if both drains, when open at the same
time, can empty the pool in 2.6 h. Confirm
algebraically.
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Slide 2- 14
Homework, Page 253
41. a. Express as a function of t the part D of the drainage
that can be done in 1 h with both drains open at the same
time.
1
1
4 1
D
D
4.75 t
19 t
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Slide 2- 15
Homework, Page 253
41. b. Find graphically the time it takes for drain B alone
to empty the pool if both drains, when open at the same
time, can empty the pool in 2.6 h. Confirm algebraically.
1
4 1 10 4 1 1 10 4
1 190 104
2.6 19 t
26 19 t
t 26 19
t
494
1 86
494
32
t
5 h 5.744h
t 494
86
43
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Slide 2- 16
Homework, Page 253
45. True – False An extraneous solution of a rational
equation is also a solution of the equation. Justify your
answer.
False. An extraneous solution is a solution of an
equation obtained by multiplying or dividing each term
of a rational equation by an expression containing a
variable. The extraneous solution is a solution of the
resulting equation, but not of the original equation.
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Slide 2- 17
Homework, Page 253
49. Which of the following are solutions of the equation
x
2
14
2
x 2 x 5 x 3 x 10
a.
b.
c.
d.
e.
x 5 or x 3
x 2 or x 5
only x 3
only x 5
There are no solutions.
14
2
x
2
x 2 x 5 x 3 x 10
x x 5 2 x 2 14
x 2 5 x 2 x 4 14
x 2 3 x 10 0
x 5 x 2 0
x 2,5
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Slide 2- 18
Homework, Page 253
Solve for x.
1
53. y 1
1 x
1
1
y 1
1 y
1 x 1 y 1
1 x
1 x
y
1 y x xy 1 x y 1 y x
y 1
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Slide 2- 19
2.8
Solving Inequalities in One Variable
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about
Polynomial Inequalities
Rational Inequalities
Other Inequalities
Applications
… and why
Designing containers as well as other types of
applications often require that an inequality be solved.
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Slide 2- 21
Polynomial Inequalities
A polynomial inequality takes the form f ( x) 0,
f ( x) 0, f ( x) 0, f ( x) 0 or f ( x) 0, where
f ( x) is a polynomial.
To solve f ( x) 0 is to find the values of x that
make f ( x) positive.
To solve f ( x) 0 is to find the values of x that
make f ( x) negative.
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Slide 2- 22
Example Finding where a Polynomial is
Zero, Positive, or Negative
Let f ( x) ( x 3)( x 4) 2 . Determine the real number
values of x that cause f ( x) to be (a) zero, (b) positive,
(c) negative.
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Slide 2- 23
Example Solving a Polynomial Inequality
Analytically
Solve x 4 x x 6 0 analytically.
3
2
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Slide 2- 24
Example Solving a Polynomial Inequality
Graphically
Solve x 6 x 2 8x graphically.
3
2
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Slide 2- 25
Example Creating a Sign Chart for a
Rational Function
x 1
Let r ( x)
. Determine the values of x that
x 3 x 1
cause r ( x) to be (a) zero, (b) undefined, (c) positive, and
(d) negative.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 26
Example Solving an Inequality Involving
a Rational Function
( x 5) 4
0
x x 3
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Slide 2- 27
Example Solving an Inequality Involving
a Radical
Solve ( x 2) x 1 0.
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Slide 2- 28
Example Solving an Inequality Involving
an Absolute Value
x3
0
x2
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Slide 2- 29
Homework
Review Section 2.8
Page 264, Exercises: 1 – 69 (EOO)
Quiz next time
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Slide 2- 30