DrDelMath

SUMMARY
Chapter 1:  Equations and Inequalities

Section 1.1: Graphs of Equations (page 78)

Definition: An equation is a mathematical statement which contains an = sign.

Definition: A number (or numbers) that makes an equation true when substituted for the variable (or variables) is called a solution of the equation.

Definition: The collection of all solutions of an equation is called the solution set of the equation.

Definition: The graph of an equation consists of all the points, and only those points, which are solutions of the equation.
An alternate, but equivalent definition is: The graph of an equation consists of all the points, and only those points, which satisfy the equation.

Equation: The equation of a circle with center at the origin and radius r is x² + y² = r²

Equation: The equation of a circle with center at the point (h, k) and radius r is (x - h)² + (y - k)² = r²

Section 1.2: Linear Equations in One Variable (page 88)

Definition: Two equations are equivalent if they have the same solution sets.

Definition: The process of finding all the solutions (the solution set) of an equation is called solving the equation.

Definition: A simplest equation is an equation which has a single variable on one side of the equal sign and a single number on the other side.

Properties of Equations:
(1) If the same expression is added to (or subtracted from) both sides of an equation the resulting equation will be equivalent to the original equation.

(2) If both sides of an equation are multiplied ( or divided) by the same non-zero real number, the resulting equation is equivalent to the original equation.

Process to solve a linear equation: Start with the original equation (the one to be solved) and use the above two properties to generate simpler equations, all equivalent to the original equation, until we arrive at the simplest equation. (p. 89)

Recall that equivalent equations have the same solution set. Therefore the simplest equation found by the above process has the same solution set as the original equation. The solution set for the simplest equation is clearly the single number on one side of the equation.

Definition: A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers with a not zero.

Definition: An x-intercept of a graph is a point where the graph intersects the x-axis.

Procedure: The x-intercepts of a graph are found by setting y = 0 and solving for x.

Definition: A y-intercept of a graph is a point where the graph intersects the y-axis.

Procedure: The y-intercepts of a graph are found by setting x = 0 and solving for y.

Section 1.3: Modeling with Linear Equations (page 97)

Percent

All percent problems are solved by referring to the basic formula that relates percentage, percent, and the base of the percentage.

    Percentage = (Percent)(Base)

There are only three percent related problems:
    (1) Given Percent and Base, calculate Percentage
   (2) Given Percent and Percentage, calculate Base
   (3) Given Percentage and Base, calculate Percent

If we let A represent Percentage, let P represent percent and B represent the Base, the the basic formula may be written as
            A = PB
which is read as follows:
   The percentage A is P percent of the base B.

Use this verbal statement and the basic formula to help translate percent problems into an equation.

Finally remember the meaning of percent is "per 100" and convert all percents to decimals.

Common Formulas (See Page 102 and back flyleaf of your textbook)

Section 1.4: Quadratic Equations (page 109)

Definition: A quadratic equation in one variable is an equation which may be written in the form
                                ax² + bx + c = 0
                     where a, b, and c are real numbers and a is not zero.

Observe that a quadratic equation is a second degree polynomial equation.

Quadratic equations may be solved by factoring and using the Zero Factor Property or by using the Quadratic formula.

Zero Factor Property:  If a and b are real numbers and ab = 0, then a = 0 or b = 0.

Quadratic Formula: The solutions of a quadratic equation ax² + bx + c = 0 are given by
                                             

Section 1.5: Complex Numbers (page 123)

See the Supplemental Material. It contains everything about complex numbers.

Section 1.6: Other Types of Equations (page 130)

Goals -- What You Should Learn 1. How to solve a polynomial equation of degree three or greater. 2. How to solve an equation involving radicals. 3. How to solve an equations involving rational expressions 4. How to solve an equation involving absolute values. 5. How to use polynomial equations to solve problems. 6. How to use equations involving radicals, rational expressions, or absolute value to solve problems.

- Vocabulary - Definitions to Memorize Always review previously memorized definitions 1. Term 2. Polynomial 3. Polynomial equation in one variable 4. Absolute Value 5. Zero Quotient Property

Links to Supplemental Material Check OnLine Drill and Practice 1. Polynomial Review 2. Review of Rational Expressions 3. Fractions of all Kinds 4. Exponents and Radicals

Definition: A term is a letter, a number, or a product of letters and numbers.

Definition: The numerical factor of a term is called its coefficient.

Definition: A polynomial is a term or a sum of terms in which all variables have whole number exponents.

Alternate Definition: A polynomial is an expression which can be written as
                             
                             where n is a whole number and each of the coefficients are real numbers.

Definition: A polynomial equation is an equation which can be written in the form
                               
                             where n is a whole number and each of the coefficients are real numbers.

Procedure: It is sometimes possible to use factoring and the Zero Factor Property to solve a polynomial equation.

Procedure: It is sometimes possible to substitute a variable for an expression in an equation so that the resulting equation is a quadratic equation. That equation may be solve by normal methods for solving quadratic equations. Those results are then substituted back to obtain solutions to the original equation.

An alternate Procedure:
1) Generate an equivalent equation with a rational expression on one side of the = sign and 0 on the other side.
This may be done using:
      a) Add any expression to both sides of the equation.
      b) Multiply both sides of the equation by any non-zero real number.
      c) Add (or subtract) rational expressions.
2) Reduce the rational expression to lowest terms.
3) Set the numerator equal to zero and solve that equaton.
4) The solution set for the original equation contains the solution set found in Step 3. Any number which yields a 0 in a denominator of the original equation cannot be a solution of the original equation.

The final step in the above process depends on:
Fundamental Principle: Zero-Quotient Property
If a and b are real numbers and the quotient = 0, then a = 0.

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.

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 symbols is positive or zero.
    Case 2: The expression inside the absolute value symbols is negative.

Every equation involving absolute value is solved by considering these two cases as in this general 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 symbols is always a linear expresion in one variable. Consequently, 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.

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.

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.

Section 1.7: Linear Inequalities in One Variable (page 141)

Properties of Inequalities
If a, b, c, and d are real numbers and x is either a variable or a constant then the following are true.

• Transitive Property

      If a < b and b < c then a < c


• Addition of Inequalities
        If a < b and c < d then a + c < b + d
  
  
• Addition of an Expression
        If a < b then a + x < b + x
  
  
• Multiplication by a Constant
        If a < b and c > 0, then ac< bc
        If a < b and c < 0, then ac > bc

Definition: An inequality is a mathematical statement which contains an inequality symbol.

Definition: A number (or numbers) that makes an inequality true when substituted for the variable (or variables) is called a solution of the inequality.

Definition: The collection of all solutions of an inequality is called the solution set of the inequality.

Definition: Two inequalities are equivalent inequalities if they have the same solution sets.

Definition: The process of finding all the solutions (the solution set) of an inequality is called solving the inequality.

Definition: A simplest inequality is an inequality which has a single variable on one side of the inequality symbol and a single number on the other side.

Definition: A linear inequality in one variable x is an inequality which can be written in the form ax + b < 0.

Properties of Inequalities:
(1) If the same expression is added to (or subtracted from) both sides of an inequality the resulting inequality will be equivalent to the original inequality.

(2) If both sides of an inequality are multiplied (or divided) by the same positive real number, the resulting inequality is equivalent to the original inequality.

(3) If both sides of an inequality are multiplied (or divided) by the same negative real number and the inequality symbol is reversed, the resulting inequality is equivalent to the original inequality.

Process to solve a linear inequality: Start with the original inequality (the one to be solved) and use the above three properties to generate simpler inequalities, all equivalent to the original inequality, until a simplest inequality is obtained.

Definition: The graph of an inequality consists of all the points, and only those points, which are solutions of the inequality.
An alternate, but equivalent definition is: The graph of an inequality consists of all the points, and only those points, which satisfy the inequality.

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.

Section 1.8: Other Types of Inequalities (page 151)

Goals -- What You Should Learn 1. How to solve a polynomial inequality. 2. How to solve a rational inequality. 3. How to use inequalities to model and solve problems.

- Vocabulary - Definitions to Memorize Always review previously memorized definitions 1. Polynomial inequality 2. Rational expression 3. Rational inequality in one variable

Links to Supplemental Material Check OnLine Drill and Practice

Definition: A polynomial inequality is an inequality which may be written in the form
                    

Procedure: To solve a polynomial inequality:
     1.  Find the Real Zeros of the polynomial.
     2.  Plot the Real Zeros on the number line to divide the number line into rays and intervals..
     3.  From each of the rays and intervals test one number in the inequality.
     4.  If the result of a test point is a true statement, the entire ray or interval is in the solution set.
     5.  If the result of a test point is a false statement, no number in the ray or interval is in the solution set.
     6.  The solutions set is the union of all the rays and intervals for which the test point yielded a true statement.

Definition: A rational expression is an expression that can be written as the quotient of two polynomials with the denominator not zero.

Alternate Definition: A rational expression is an expression that can be written as
                          
                        with the denominator not zero.

Definition: A rational inequality is an inequality which may be written in the form

Procedure: To solve a polynomial inequality:
     1.  Find the Real Zeros of the numerator and find the Real Zeros of the denominator.
     2.  Plot these Real Zeros on the number line to divide the number line into rays and intervals..
     3.  From each of the rays and intervals test one number in the inequality.
     4.  If the result of a test point is a true statement, the entire ray or interval is in the solution set.
     5.  If the result of a test point is a false statement, no number in the ray or interval is in the solution set.
     6.  The solutions set is the union of all the rays and intervals for which the test point yielded a true statement.