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 .

 

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Test Out Exams will be administered from 2:00 to 5:00 pm in UGLC 116 on the following date:

 

   01/17/03 - 2:00 p.m. to 5:00 p.m.    UGLC 116
   02/11/03 -
2:00 p.m. to 5:00 p.m.    PSCI 115
   03/7/03 -
2:00 p.m. to 5:00 p.m.    PSCI 115
   04/11/03 -
2:00 p.m. to 5:00 p.m.    UGLC 116
   05/9/03 -
2:00 p.m. to 5:00 p.m.    UGLC 116
   06/02/03 -
2:00 p.m. to 5:00 p.m.    UGLC 116

 

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.