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DrDelMath

PRECALCULUS by Barnett/Ziegler/Byleen
SUMMARY

Chapter 2: Linear and Quadratic Functions

Section 2.1: Linear Functions (page 128)

Everything about linear functions is contained in the Special Topic titled Linear Functions listed under Special Topics. It is strongly recommented that you study that document.

A few other miscellaneous formulas related to lines are presented in this section of the text and are repeated here

Recommended Exercises Page 146.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
19-28, 29-40, 41-48, 53-70.

Section 2.2: Linear Functions and Models (page 151)

Read/study the Text and each of the following documents.
Document 1 -- Constructing a mathematical model
Document 2 -- Mixture Problems
Document 3 -- Word Problems -- miscellaneous examples.

Recommended Exercises Page 167.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
1, 3, 13-18, 19-30, 31-46, 47-56.

Section 2.3: Quadratic Functions (page 172)

Everything about quadratic functions is contained in the Special Topic titled Quadratic Functions listed under Special Topics. It is strongly recommented that you study that document.

Recommended Exercises Page 186.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
1, 2, 3, 41-48, 49, 51, 53.

Section 2.4: omitted --- Complex Numbers (page 190)

Be sure to study and understand Figure 2 on Page 194.

Be sure to study and understand the "Caution" on Page 202.

Recommended Exercises Page 204.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
1, 2, 3, 4, 5, 6, 7-12, 13-46.

Section 2.5: Quadratic Functions and Models (Page 206)

Be sure you can solve quadratic equations using factoring and the Zero Factor Property or with the Quadratic Formula.

Recommended Exercises Page 221.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
1-6, 19-26, 89 1, 3, 4, 5, 6, 7-16, 29-38, 51-60, 65-80.

Section 2.6: Additional Equation Solving Techniques (Page 226)

RADICAL EQUATIONS
  Illustration
RATIONAL EQUATIONS

  Illustration

Recommended Exercises Page 239.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
1-8, 19, 20, 21, 22, 27, 32, 57, 58.

Section 2.7: Solving Inequalities (page 210)

Properties of Inequalities:

The remaining material is presented as a review of absolute value. It is more extensive that the text coverage of this topic and is presented here for you convenience.

Other Inequalities

Click on the following link for a discussion of other kinds of inequalities.

   Law of Trichotomy and Boundary Equations

Absolute Value Equations and Inequalities

Solving Equations Involving Absolute Value

The basis for solving all equations involving absolute values is the definition of absolute value.

Notice the precise definition of absolute value has two cases:
    Case 1: The expression inside the absolute value symbol is positive or zero.
    Case 2: The expression inside the absolute value symbol is negative.

Procedure: To solve equations involving absolute values of variable expressions, it is necessary to solve the two equations which naturally result from the definition of absolute value. The solution set contains the solution set of the original equation. Therefore an integral part of the solution process is to test each of the possible solutions in the original equation.

Every equation involving absolute value is solved by considering the two cases as in this general example:
Generic Example: To solve an equation involving |something|, two cases must be considered. The two cases arise from the definition of absolute value. Therefore to solve any equation involving |something|, we consider:
    Case 1: The equation that results from relacing |something| with (something)
    Case 2: The equation that results from relacing |something| with -(something).

The following two properties of equations are also important tools when solving equations involving absolute value.

These two properties might be used to:
    Simplify the equation before considering the two cases.
    Solve the equations in Case 1 and Case 2.

In this section of the textbook the expresion inside the absolute value symbol is always a linear expresion in one variable. All of these equations can be solved using the two cases that come from the definition of absolute value together with the two properties for generating equivalent equations.

Solving an equation of the type |ax + b| = c.
     Consider the two cases.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = c.
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = c.
In each case, the two properties for generating equivalent equations may be used to find a simplest equation.
The union of the solution sets for the two cases is the solution set for the original equation.

Solving an equation of the type |ax + b| + c = d.
     Add the expression -c to both sides to obtain the equivalent equation |ax + b| = d - c.
     Consider the two cases.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = d - c.
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = d - c.
In each case, the two properties for generating equivalent equations may be used to find a simplest equation.
The union of the solution sets for the two cases is the solution set for the original equation.

Solving an equation of the type |ax + b| = |cx + d|.
     Consider the two cases.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = cx + d.
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = cx + d.
In each case, the two properties for generating equivalent equations may be used to find a simplest equation.
The union of the solution sets for the two cases is the solution set for the original equation.

It may seem that there should be four cases in this last type of equation, but there are duplications as shown here.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = |cx + d|.
         Case 1A: (when cx + d is positive or zero) Solve the equation ax + b = cx + d.
         Case 1B: (when cx + d is negative) Solve the equation ax + b = -(cx + d).
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = |cx + d|.
         Case 2A: (when cx + d is positive or zero) Solve the equation -(ax + b) = cx + d.
         Case 2B: (when cx + d is negative) Solve the equation -(ax + b) = -(cx + d).
If both sides of the equation in Case 2B is multiplied by -1, the equivalent equation obtained is identical to the equation in Case 1A.
If both sides of the equation in Case 2A is multiplied by -1, the equivalent equation obtained is identical to the equation in Case 1B.
For that reason only two cases are necessary.

Solving an equation of the type |ax + b| = c where c is a negative number.
Simply observe that absolute value is always non-negative (positive or zero) and therefore this type of equation does not have a solution. It solution set is the empty set.

Solving an equation of the type |ax + b| = 0.
The only way an absolute value can be zero is for the expression inside the absolute value symbol to be zero. So only Case 1 needs to be considered. No harm is done if Case 2 is considered, because in this type of equation it will always give the same solution set as Case 1.

Definition: Two inequalities joined by the words AND or OR are called compound inequalities.
FACT: The solution set of a compound inequality formed by the word AND is the intersection of the solution sets of the two inequalities.
FACT: The solution set of a compound inequality formed by the word OR is the union of the solution sets of the two inequalities.

Warning: When using the compact form of a compound inequality such as a < x < b, be careful to write only meaningful statements. The inequalities must both "point" in the same direction. The end expressions must be related as indicated by the inequality symbols. For example, in a < x < b, it must be true that a < b.

FACT: If k is a positive number and X is either a single variable or a variable expression, then the inequality
                      | X | < k   is equivalent to   - k < X < k
Note this is a compound inequality formed with the word AND. Therefore the solution set is the intersection of the solution sets of the two individual inequalities.

FACT: If k is a positive number and X is either a single variable or a variable expression, then the inequality
                       | X | > k   is equivalent to   X < - k OR X > k
Note this is a compound inequality formed with the word OR. Therefore the solution set is the union of the solution sets of the two individual inequalities.

Recommended Exercises Page 254.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
1, 2, 3, 4, 25-38, 39-64.