Section 11.3
1. a. Windows may vary; Xmin = 0, Xmax= 9.4,
Ymin = 0, Ymax = 1.
b. The intersection ofYI and Y2 is
approximately (6, .66666667). This indicates
that the solution to the equation is x = 6, which
alZreeswith the result in ExamDle3.
3. YI = l/(X-2) + 3; Y2 = 7
Windows may vary; Xmin = -9.4, Xmax = 9.4,
Ymin = -10, Ymax = 10
After pressing 2nd CALC intersect, you may need
to move the cursor right in order to start on the
rimt branch of the hVDerbola.
ut
InhrslQ:cti<afl
M=2:.2:5
IY:?
The solution is x = 2.25.
@ Houghton MifflinCompany. All rights reserved.
-
Section 11.3
5. Let x be represent the time (in hrs) for the intake
pipe to fill the pool working alone. Then x + 3
represents the time (in hours) for the outtake pipe
to drain the pool working alone. Following the
same method of reasoning and solution in
Example 4, in this problem we have
I I
----x
1*(x+3)
x*(x+3)
6
I
(x+3)*x
x+3-x
6
I
x(x+3)
6
3
I
x2 +3x -'6
x2 + 3x = 18
x2
x+6=0
+ 3x -18
=0
(x+6)(x-3)=0
or x-3=0
x= -6 or x=3
Since x represents time, we ignore the negative
solution. Thus it takes 3 hours to fill the pool with
the outtake pipe shut.
7.
x-2
6(x-2)=20
6x-12=20
6x = 32
32 16
x=-=6 3
3*x
-+
(x-l)*x
~+
x(x-I)
3
8
-+-=3
x-I x
8*(x-l) -3
x*(x-I)
8x-8 =3
x(x-I)
3x + 8x - 8 _ 3
x(x -I)
llx-8=3
x2-x
llx-8 = 3(x2-x)
Ilx-8 = 3x2-3x
0= 3x2-14x+ 8
3x-2=0
Work
Time
Joan
1 car
4 hours
t
Partner
I car
6 hours
t car/hour
Together
I car
x hours
1.
x car/hour
0= (3x- 2)(x-4)
or x-4=0
2
x=- or x=4
3
@ Houghton MifflinCompany. All rights reserved.
Rate
car/hour
Since Joan and her partner are working together,
we add their work rates:
I I I
-+-=-
4 6
1*6 1*4
-+-=4*6 6*4
6+4
---
x
I
x
24 x
10
24 x
lOx = 24
x
6=~
9.
11. Let x represent the time (in hours) that it takes for
Joan and her partner to detail a car working
together. Recognize that the time should be less
than 4 hrs (the time it takes Joan to do the job by
herself) because she has her partner's help.
Construct a table as follows:
I
x+3
I--*x
159
= 2.4
hrs
= 2 hrs
24 min
It takesJoanandherpartner2.4hoursor 2 hours
24 minutesto detaila car together. Thismakes
sensebecauseworkingtogetherwill requireless
timethan eitherpersonworkingalone.
13. Letx be the speedof the rivercurrent(in mls).
a. Sincethe teamcanrow 80mls in stillwater,
then they can row (80 - x) mls going
upstream.
distance= rate * time
3000= (80- x)50
60=80-x
-20 = -x
20=x
The speedof the rivercurrentis 20 mls.
b. Theteamrowsat a rate of
80 mls+ 20 mls = 100mls downstream.The
totaltravelingdistanceis still3000m.
.
distance
tune =rate
3000
100
=30
Therowingteamtravelsdownstreamin 30
seconds.
160
Chapter 11: Rational Expressions and Functions
15. a. Windows may vary: Xmin = "9.4,Xmax = 9.4,
Ymin = "10, Ymax = 15
25. a.
x2 + 5x = -6
x2 + 5x + 6 = 0
(x + 3)(x+ 2) = 0
x+3=0 or x+2=0
x = -3 or x = -2
b.
4(10)' = 400
lOx= 100
log(10X)= log(100)
No, the graph ofY1= (7X + 3)/X has a
horizontal asymptote at Y = 7. As X assumes
very large values (negative or positive) the
value of Y will approach 7, but never equal 7.
b. 7x + 3 = 7
x
7x+3=7x
3=0
But 3 * 0
x = log(102)
x=2
c. x = 10g(1O)= log(101)= 1
d.
3 4 =-48
-x
5
5
48
x =-*- 5
5 3
x4 = 16
4
c. Either by trying to solve the equation
(X4Y'4
7x + 3 =7 by graphing or with algebra, we
x
fmd the equation has no solution.
Skills and Review 11.3
l*x
121
-+-=2 x
2*2
-+-=2*x x*2
-x+4
2x
y1
=
(x +x 1)
b. Vertical asymptote: y-axis (x = 0);
horizontal asymptote: y = 1
19.' ~-'-
3x2 =~*
7-x
X- 7 7- x x - 7 3x2
7x-x2
- 3x2(x-7)
_ -lx(x -7)
- 3x2(x-7)
-1
3x
power 16 = 2'
x=:l:2
e.
17. a.
= :I:(16i/4
X = :1:(24)114Use a familiar
x
1
x
x
x(x + 4)
= 2x
x2 + 4x
= 2x
x2 + 2x
=0
x(x+2)=0
x=00rx+2=0
x=O orx=-2
x = 0 is an extraneoussolutionbecausewhen0
is substitutedforx intothe originalequationit
givesundefinedterms. Thereforethe only
solution is x
= -2.