Transcript Slide 1

Homework, Page 253
Solve the equation algebraically. Support your answer numerically
and identify any extraneous solutions.
1.
x2 x5 1


3
3
3
x2 x5 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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 6
Homework, Page 253
Solve the equation.
25.
x2  2 x  1
0
x5
x2  2 x  1
2
2
 0  x  2 x  1  0   x  1  0
x5
x  1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 7
Homework, Page 253
Solve the equation.
29.
5
x  8
x
2
x  3.100,0.661, 2.439
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 23
Example Solving a Polynomial Inequality
Analytically
Solve x  4 x  x  6  0 analytically.
3
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 24
Example Solving a Polynomial Inequality
Graphically
Solve x  6 x  2  8x graphically.
3
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
x3
0
x2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 29
Homework



Review Section 2.8
Page 264, Exercises: 1 – 69 (EOO)
Quiz next time
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Slide 2- 30