Modular Precalculus On-Line
Solutions -
Assignment 11 - Part 4
Part
1 includes the material in Appendices A, B, C, D, and E and Chapters 1, 2, and
3. You may ask questions via email to
the following peer facilitators: Jonathan at szarzynski@hotmail.com, Erika at akire1@hotmail.com,
Denise at dminortx@yahoo.com, Ryan at averymaxfield@hotmail.com, Claudia at
claudiav@miners.utep.edu. You may also
post questions at the sosmath.com cyberboard at http://www.sosmath.com/cgi-bin/UBB/ultimaatebb.cgi
You
may also visit the peer facilitators at their office in Bell Hall 130 during
their office hours. If you decide to
post your name and email address on the online roster, you can also email other
members of the online class for some electronic collaboration. If you want your
name and email address posted to the roster, let Jonathan or Erika know.
The
amount of time you take for an online assignment or online exam is up to
you. We are trying to prepare you for
the Part 4 Test Out Exam. You must score at least 70 to pass Part 4.
Assignment
11:
Read
Chapter 7 - Section 3.
Answer
the following questions, and work the following problems:
Questions:
Problem 1:
What is the difference between an equation and an
identity?
Solutions 1:
An identity is an equation that is true for all values of the variable(s). An equation is true for a finite number of values of the variable(s).
For example, 
is true for every
value of x and is therefore an identity.
For example, 
is true only when 
.
Problem 2:
How can you illustrate that 
is equivalent to 
graphically?
Solutions 2:
Graph and graph on the same graph
display and you will just one graph.
Problem 3:
Is 
an equation or an
identity? Why?
Solutions 3:
If you graph 
and 
on the same
graphically display, you will see two graphs that intersect at finite
points. This indicates that the above
equation is not an identity.
Problem 4:
If the expression 
is a valid expression,
what are the restrictions on the domain?
Solutions 4:
We must eliminate all those values of x such that 
.
when
and when 
. Therefore, these
values must be eliminated from the domain.
Problem 5:
Solve for the exact value(s) of t in
.
Solution 5:
Rewrite an 
and solve.
when
.
is
impossible because there is no t such that 
.
Problem 6:
Solve for the exact value(s) of t in
.
Solution 6:
Only 
works. Why?
The values of the sine function are restricted such that
. 
is
outside that range.
Problem 7:
Solve
for the exact value(s) of t in 
.
Solution 7:
when
.
when
Problem 8:
Solve
for the exact value(s) of t in 
Solution 8:
Solve this equation
graphically.
Problem 9:
Solve
for the exact value(s) of t in 
.
Solution 9:
Solve this equation graphically.
Problem 10:
Solve
for the exact value(s) of t 
.
Solution 10:
Solve this equation
graphically.
Problem 11:
Why
is the inequality 
a false statement?
Solution 11:
It
is always positive.
Problem 12:
What
is the domain of the function 
? Why doesn’t 
exist?
Solution 12:
The
domain is the set of real number x, where 
. The critical values
are those values of x where 
. The critical values
divide the number line into subintervals, and the solution is one or more of
these intervals.
are
the critical values
The
test intervals are 
.
-1
is in the first interval and 
. Therefore all the
real numbers in the first interval is a set of solutions.
0
is in the second interval and 
which is not greater
than 0. Therefore no number in the
second interval is a solution.
10
is in the third interval and 
. Therefore all the
real numbers in the third interval is also a set of solutions.
The
solution set is the set of real numbers in the interval 
and all the real numbers
in the interval 
.
Problem 13:
Solve 
.
Solution 13:
The
value of a fraction equals zero when the numerator equals zero. The solution is 
.
Problem 14:
Add
the following fractions and simplify.
Solution 14:
Problems:
Problem 1. Problem # 3 in
Section 7.3
Solution 1:
Problem 2. Problem # 13 in
Section 7.3
Solution 2:
The graphs of the
left and right side are the not the same, therefore this equation is not an
identity.
The solutions are 0 and 8.
Problem 3. Problem # 15 in
Section 7.3
Solution 3:
is
not an identity.
Problem 4. Problem # 18 in
Section 7.3
Solution 4:
is
an identity.
Problem 5. Problem # 23 in Section 7.3
Solution 5:
a) 
and
On
the interval 
, 
and 
b) 
and
and
Problem 6. Problem # 28 in Section 7.3
Solution 6:
If 
, then 
, and 
, and 
.
Problem 7. Work all the odds in Appendix H.
Solution 7:
H – 1: 
The
critical value is 7. The test intervals
are 
and 
.
The number 0 is located in the
interval and 
which is not greater
than zero. The interval is not a
solution.
The number 10 is located in the
interval and 
which is greater than
zero. The interval is a solution.
The
solution is the set of real numbers in the interval .
H – 2: 
The
critical value is 16. The test intervals
are 
and 
.
The number 0 is located in the
interval and 
which is
not greater than 4. The interval is not
a solution.
The number 100 is located in the
interval and 
which is
greater than 4. The interval is a
solution.
The
solution is the set of real numbers in the interval .
H – 3: 
which can be written as 
and 
The
critical values are 
16. The test intervals are 
, 
and 
.
The number -10 is located in the
interval and 
which is
not less than 0. The interval is not a
solution.
The number 0 is located in the
interval and 
which is less
than 0. The interval is a solution.
The number 10 is located in the
interval and 
which is
not less than 0. The interval is not a
solution.
The
solutions is the set of real numbers in the interval .
H – 4: 
The
critical value is 5. The test intervals
are 
and .
The number 0 is located in the
interval and 
which is
not less than 2. The interval is not a
solution.
The number 10 is located in the
interval and 
which is greater
than 2.
The
interval is a solution.
The
solution is the set of real numbers in the interval .
H – 5: 
can be written as 
which is
always true if 
The
solution is the set of real numbers in the interval 
.
H – 6: 
which can be written as 
.
H – 7: 
can be written as 
The
critical values are 
. The test intervals are 
, 
and 
.
The number -10 is located in the
interval and 
which is greater
than 16. The interval is a solution.
The number 0 is located in the
interval and 
which is not
greater than 16. The interval is not a
solution.
The number 10 is located in the
interval and 
which is greater
than 16. The interval is a solution.
The
solutions is the set of real numbers in the intervals and .
H – 8: 
is true for all values of x. The
solution is the set of real numbers.
H – 9: 
when 
The
solutions is the set of real numbers in the intervals .
H – 10: 
is true for all real values of x.
The
solution is the set of real numbers.
H – 11: 
can be written as 
which is defined to be
the set of real numbers in the interval 
.
H – 12: 
There
is no real number x, such that .
H –13: 
The
denominator is always positive and the numerator is positive when x is greater
than 5.
The
solutions is the set of real numbers in the interval .
H – 14: 
The expression 
is always
positive. Since the product of a
positive number and a negative number is a negative number, then 
or 
The
solutions is the set of real numbers in the intervals 
.
H – 15: 
The expression 
is always positive. Since the quotient of a positive number and a
negative number is a negative number, then 
or 
The
solutions is the set of real numbers in the intervals 
.
H – 16:
The narrative phrases “the x-values which are less
than 0.001” can written in symbolic form as 
.
H – 17:
The narrative phrases “the y-values between -1 and
1” can written in symbolic form as 
.
H – 18:
The narrative phrases “all p x-values except 5” can
written in symbolic form as 
.
H – 19:
The narrative phrases “all positive values of x” can
written in symbolic form as 
.
H – 20:
The narrative phrases “all the r-values which are
not negative” can written in symbolic form as 
.
H – 21:
The narrative phrases “the t-values during or after
the year 1995 
.
H – 22: 
can be simplified to 
The solution is the set of real numbers greater than
4. When graph this inequality, you would
draw a dotted vertical line at 
. The entire space to
the right of that line is the solutions set.
H – 23: 
can be simplified to 
The solution is the set of real numbers greater than
2. When graph this inequality, you would
draw a dotted vertical line at 
. The entire space to
the right of that line is the solutions set.
H – 24: 
can be simplified to 
The solution is the set of real numbers less than
-4. When graph this inequality, you
would draw a dotted vertical line at 
. The entire space to
the left of that line is the solutions set.
H – 25: 
can be simplified to 
and again 
.
The solution is the set of real numbers less than
-4. When graph this inequality, you
would draw a dotted vertical line at 
. The entire space to
the right of that line is the solutions set.
H – 26:
The
critical values are 2 and 3 . The test intervals are 
, 
and 
.
The number -10 is located in the
interval and 
which is greater than
0. The interval is a solution.
The number 2.5 is located in the
interval and 
which is not greater
than 0. The interval is not a solution.
The number 10 is located in the
interval and 
which is greater than
0. The interval is a solution.
The
solutions is the set of real numbers in the intervals and .
When graph the function 
, it is a parabola that opens up and crosses the x-axis at
the points 2 and 3. Note that to the
left of 2 and the right of 3 the graph is above the x-axis (function is greater
than 0).
H – 27:
The
critical values are -1 and 8 . The test intervals are 
, 
and 
.
The number -10 is located in the
interval and 
which is not less than
8. The interval is a not solution.
The number 0 is located in the
interval and 
which is less than 8. The interval is a solution.
The number 10 is located in the
interval and 
which is not less 8. The interval is not a solution.
The
solutions is the set of real numbers in the interval .
When graph the function 
, it is a parabola that opens up and crosses the x-axis at
the points -1 and 8. Note that between 
and 
, the graph falls below the x-axis (function is less than 0).
H – 28:
The solution is the set of real numbers greater than
3.
H – 29:
The critical values are those values that make the
numerator zero or the denominator zero.
Therefore, the critical values are 
and 5.
The
test intervals are , 
and 
.
The number 0 is located in the
interval and 
which is not greater
than 3. The interval is a not solution.
The number 5.5 is located in the
interval and 
which is greater than
3. The interval is a solution.
The number 6 is located in the
interval and 
which is not greater
than 3. The interval is a not solution.
The
solutions is the set of real numbers in the interval .
H – 30:
This is true for all values of x such that or 
.
H – 31:
This is true for all values of y such that 
or 
If , then 
, If , then 
.
The
solutions is the set of real numbers in the intervals and 
.
====
===
Test
Out Exams will be administered from
01/17/03 -
02/11/03 -
03/7/03 -
04/11/03 -
05/9/03 -
06/02/03 -
The
exams may be multiple choice or they may be essay (all work shown).
If
you would like to go to the Assignment Page of Part 1, click on Part
1
If
you would like to go to Part 2, click on Part
2
If
you would like to go to Part 3, click on Part
3
If
you would like to go to Part 4, click on Part
4
If
you have any questions, contact Dr. Nancy Marcus at nancymarcus@utep.edu, call her at
747-6750, or drop by and see her in Bell Hall 100.