Advanced Algebra

Chapter 5 - Note Taking Guidelines

5.1 Inequalities and Compound Sentences

1. When is a mathematical sentence called an inequality?

2. What is an open sentence?

3. How do we graph inequalities on a number line?

4. When graphing an inequality when do we use a shaded circle?

5. When graphing an inequality when do we use a circle?

6. What do we call all the numbers that satisfy an inequality?

7. What is a compound sentence?

8. Describe the solution set of a compound sentence which uses the word and to connect two inequalities.

9. Study example 1

10. Now try the following problem:

a. As of 1994, recorded temperatures in the state of Michigan have ranged from a low of -51o F in Vanderbilt (1934) to a high of 112o in Mio (1936). Graph the range of recorded temperatures.

11. How do we read BA ∩ ?

12. Describe the solution set of a compound sentence which uses the word or to connect two inequalities.

13. How do we read BA ∪ ?

14. Study example 2

15. Now try the following problem:

a. People from age 16 to 65 can give blood at a Blood Bank.

i. Write a compound inequality to describe the possible ages A at which blood can be given.

ii. Graph the inequality on a number line.

16. Properties of Inequality:

17. Study example 3

18. Now try the following problem:

a. Solve 2643 ≤−x and graph the solution set

19. While solving inequalities when are we required to reverse the relation?

20. Study example 4

21. Now try the following problem:

a. An airplane flying at 34,000 feet descends at the rate of 2500 feet per minute. After how many minutes will the plane be below 20,000 feet?

22. Now try the following problem:

a. Graph { }π≥< xorxx 3:

Summarize what you learned in sections 5.1:

5.2 Solving Systems Using Tables or Graphs

1. What is a system?

2. Give an example of a system which uses a brace.

3. What is the solution set for a system?

4. How can we find the solution of a system?

5. Study example 1

6. Now try the following problem:

a. Solve the system

+=

+=

53

84

xy

xy

7. What do we know about the solution to a system found using an automatic grapher?

8. Study example 2

9. Now try the following problem:

a. Fred wants to enclose 400 m2 rectangular garden with 70 m of fencing. To do this, Fred must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

10. Now try the following problem:

a. Solve the system

+−=

=

13

5

xy

xyby making a

table to estimate the solution.

b. Graph the system

11. Now try the following problem:

a. Use an automatic grapher to solve the

system

=

=

xy

xy

3

3

Summarize what you learned in sections 5.2:

5.3 Solving Systems Using Substitution

1. Why do we need algebraic techniques to solve systems?

2. The substitution method for solving systems of equations uses what property of equality?

3. Study example 1

4. Now try the following problem:

a. Solve the system

+=

=+

2

6

xy

yx

5. Study example 2

6. Now try the following problem:

a. The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of three equations, and solve the system to find how many tickets of each type were printed

7. We have been working with systems of linear equations but the substitution method can also be used to solve what other type of system of equations?

8. Study example 3

9. Now try the following problem:

a. Solve the system

=

=

36

4

xy

xy

10. What is a consistent system of equations?

11. What is an inconsistent system of equations?

12. How do we know if a system is inconsistent?

13. Study example 4

14. Now try the following problem:

a. Describe the graph and the solution

=+

−=

73

34

yx

xy

15. Now try the following problem:

a. Describe the graph and the solution

=

=

2

2

63

2

xy

xy

16. Study example 5

17. Now try the following problem:

a. Determine whether the system is consistent or inconsistent. If it is consistent, tell how many solutions the system has.

i.

=

=

0

122

y

xy

ii.

=

=

xy

xy

3

2

iii.

−=−

−=−

624

32

yx

yx

iv.

=−

=−

642

42

yx

yx

18. What indicates that a system has infinite solutions?

19. How do we write the answer to a system that has infinite solutions?

20. Now try the following problem:

a. Solve the system

−=−

−=−

783

92

yx

yx

Summarize what you learned in sections 5.3:

5.4 Solving Systems Using Linear combinations

1. If the equations in a system are in standard form what method should we use to solve the system?

2. Study example 1

3. Now try the following problem:

a. Solve the system

=+

=−

756

1153

yx

yx

4. Study example 2

5. Now try the following problem:

a. A school rents its swimming pool for private parties. There is a fixed cost f which remains the same regardless of the number of guests. There is also a charge per guest, g. For a party of 40 people, the total charge is $ 230 and for a party of 125 people, the total charge is $ 400. Find the fixed cost and the charge per guest.

6. How do we use the linear combination method when there are three equations in standard form?

7. Study example 3

8. Now try the following problem:

a. Solve the system

=−+

=+−

=++

2637

3532

22543

cba

cba

cba

9. What does it mean when the linear combination method results in a sentence like 0 = 12?

10. What does it mean when the linear combination method results in a sentence like 12 = 12?

Summarize what you learned in sections 5.4:

5.7 Graphing Inequalities in the Coordinate

Plane

1. What does a line graphed in a plane do to the plane?

2. What is the line called?

3. Which half-plane does the boundary line belong to?

4. Study example 1

5. Now try the following problem:

a. Graph the solutions to y < 3 i. On a number line.

ii. In the coordinate plane.

6. When we consider only numbers that satisfy an inequality in one variable, what is the graph of all solutions?

7. When we consider ordered pairs that satisfy an inequality in one variable, what is the graph of all solutions?

8. Study example 2

9. Now try the following problem:

a. The price of a general-admission ticket at the ball park depends on a person’s age rather than height. A person 12 years old and older pays the adult price for a ticket. Graph the constraint “A person 12 years old and older pays the adult rate for a general-admission ticket, regardless of height.” Let H = the height of a person A = the person’s age

10. Study example 3

11. Now try the following problem:

a. Graph the linear inequality 53

4+≥ xy

12. What is the graph of the solution to an inequality?

13. How do we know if the boundary line should be included (solid) or not included (dashed)?

14. Define what is meant by a lattice of points.

15. Study example 4

16. Now try the following problem:

a. Leo has at most $1.50 in his pocket. i. Draw a graph showing all possible

combinations of dimes and quarters that Leo could have.

ii. How many possible combinations of dimes and quarters could Leo have?

17. Summary of steps for graphing a linear inequality in the coordinate plane:

Summarize what you learned in sections 5.7:

5.8 Systems of Linear Inequalities 1. Describe the solution to a system of linear

inequalities.

2. What is the set of solutions to a system of linear inequalities called?

3. What are the boundaries of the feasibility region for a system of linear inequalities?

4. What are the intersections of the boundaries called?

5. Study example 1

6. Now try the following problem:

a. Graph the feasible set for the system below and give the coordinates of its vertices.

≥

≥

≤+

≤+

0

0

122

122

y

x

yx

yx

7. Study example 1

8. Now try the following problem:

a. The Random Company manufactures two products, Zeta and Beta. Each product must pass through two processing operations. Zeta requires one hour for each process, and Beta requires 2 hours for process I and 3 hours for process II. Process I has a total capacity of 1000 hours per day and process II has a total capacity for 1275 hours per day.

i. Identify the variables

ii. Make a table to organize the

information.

iii. Write sentences to model each limiting resource in the process.

iv. Graph the feasibility region for the system

v. Identify the vertices of the feasible region

Summarize what you learned in sections 5.8:

5.9 Linear Programming I 1. The process which maximizes or minimizes a linear

combination using a set of constraints and linear inequalities is called…

2. Linear Programming Theorem:

3. Study example 1

4. Now try the following problem:

a. An electronics firm makes two types of electronics: The iPod Touch and the iPhone (16GB). The company has the equipment to produce at most 600 iPods or 525 iPhones. It takes 30 hours of labor to produce an iPod Touch and 40 hours of labor to produce an iPhone. The company has up to 24,000 labor hours available each month for electronics production. If the profit gained on each iPod Touch is $18 and on each iPhone is $24, find the number of each kind the firm should manufacture to gain the maximum profit each month. .

Summarize what you learned in sections 5.9:

5.10 Linear Programming II 1. What are the steps for solving an optimization

problem using linear programming?

2. Study example 1

3. Now try the following problem:

a. For his evening meal, Karl plans to eat fish sticks and mashed potatoes. Each ounce of fish sticks contains 5 grams of protein and 0.1 mg of iron. Each cup of mashed potatoes contains 4 grams of protein and 0.8 mg of iron. Karl wants to have at least 28 grams of protein and 2 mg of iron in his evening meal. If each ounce of fish sticks costs 30 cents and each cup of potatoes costs 69 cents, how much of each should Karl eat to satisfy his dietary requirements while minimizing the cost?

Summarize what you learned in sections 5.10: