Equations and Inequalities with Absolute Value
Def. (1) A relation is a set of ordered
pairs. (2) A relation is a rule of
correspondence between two sets.
EX1. { (Weevil, UAM), (Bulldog, Star City), (Bulldog, LA Tech)}
EX2. To each UAM student record his/her GPA.
Def. (1) A function is a relation in which
no two ordered pairs having the
same first elements have different second elements. (2) A function
is a relation in which to each element in the first set (the domain) there
corresponds exactly one element in the second set (the range).
Question: Is the relation in EX1 a function? Why or why
not?
Question: Is the relation in EX2 a function? Why or why
not?
Activity: Give a specific example of a relation that is
not a function.
Activity: Give a specific example of a relation that is a
function.
Activity: Give a specific example of a function that is
not a relation.
Sometimes a function is represented using functional
notation. The symbol
f(x) is read “f of x” and indicates that the function f is written in terms of
the
variable x. [Note: f(x) does not mean f times x.] f(x) represents the real
number in the range of the function f corresponding to the domain value x.
EX3: If f(x) = 2x^{2}  x + 3 find
f(2)
f(3)
f(t)
INTERVAL NOTATION:
SYMBOLIC ALGEBRAIC FORM GRAPH
FORM
[a,b]
(a,b)
[a,b)
[a, ∞)
(a, ∞)
(∞, a]
(∞, a)
(∞, ∞)
The absolute value of a number is
A) always positive
B) always an integer
C) never a fraction
D) never negative
Interpret the following geometrically. Represent your
solution on the number
line and in set notation.
EX4:  x  = 5
EX5:  x  = 3
EX6:  x  1  = 4
EX7:  x+ 1  = 4
EX8. Write an absolute value equation corresponding to the following graph
The geometric meaning of  x  a  = b where b > 0 is
The algebraic solution of absolute value
equations of the form  x  a  = b is
EX9: Solve and graph.  2x  3  = 7
EX10: Solve and graph  x  7  < 2
EX11: Solve and graph  x + 5  < 3
The geometric meaning of absolute value
inequalities of the form
 x  a  < b is
The algebraic solution of absolute value
inequalities of the form
 x  a  < b is
EX12: Solve and graph  3  2x ≤ 5.
EX13: What set of numbers is described by  x  1  > 4?
EX14: Solve and graph  x  3  ≤ 2
EX15: Solve and graph  x + 4  > 3
The geometric meaning of absolute value
inequalities of the form
 x  a  > b is
The algebraic solution of absolute value
inequalities of the form
 x  a  > b is
EX16: Solve and graph  2x  5  ≥ 8
EX17: Graph y =  x 
EX18: Graph y =  x  2  + 3
EX19: Solve  x  3  =  2x + 5  and check your answer(s).
EX20: A car CD player has an operating temperature of  t
 40°  ≤ 80°
where t is the temperature in degrees Fahrenheit. Solve for t and
give the range of operating temperatures for the CD player.
Chapter 5, Sect. 1: Polynomials and Polynomial
Functions
Def. Algebraic terms  expressions that contain constants
and/or variables
Examples:
Def. Polynomial – the sum of one or more algebraic terms
whose variables
have whole number exponents
Examples:
Def. Monomial –
Examples:
Def. Binomial –
Examples:
Def. Trinomial –
Examples
Def. If a ≠ 0, the degree of a^n is n. The degree of a
monomial containing
several variables is the sum of the exponents of those variables. The
degree of a polynomial is equal to the degree of its highest degree
term.
Examples:
Note 1: The degree of a nonzero constant is 0.
Explanation:
Note 2: The degree of 0 is indeterminate.
Explanation.
Graphs of Special Functions:
Linear Function 
Squaring Function 
EX. y = 2x + 1 
EX. y = x^{2} 
Domain
Range 
Domain
Range 
Cubing Function 
Absolute Value Function 
EX. y = x^{3} 
EX. y = x 
Domain
Range 
Domain
Range 
EX1: Write x^{2} + 3x^{5}  7x + 3x^{3} in (A) descending order and
(B) in ascending order
Descending order:
Ascending order:
EX2: If P(x) = 3x^{2}  2x + 1, find P(4), P(½), and P(n).
EX3: The height, h, in feet, of a ball shot straight up
with an initial velocity
of 64 feet per second is given by the polynomial function
h(t) = 16t^{2} + 64t
Find the height of the ball in (A)3 seconds and (B)in 1.78 seconds.
EX4: The number of feet that a car travels before stopping
depends on the
driver’s reaction time and braking distance. For a certain driver, the
stopping distance is given by d(v) = 0.04v^{2} + 0.93v where v is
the speed of the car in mph. Find the stopping distance if the speed of
the car is (A) 35 mph, (B) 55 mph, and (C) 70.86 mph.
