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1.2 – Open Sentences and Graphs
Vocabulary:
An
is a number, a variable, or a sum, difference, product, or quotient that
contains one or more variables.
Examples:
A
is a symbol, usually a letter, that represents any of the members of a
specified set. This set is called the
of the variable, and its members are called the
of the variable.
A
is a group of symbols that states a relationship
between two mathematical expressions. These can be either true or false.
Examples:
An
is a mathematical sentence that contains one or more
variables. An open sentence cannot be determined true or false without knowing what value the variable
represents.
Examples:
The values of the variable that make an open sentence true are called the
open sentence.
The
of the
is the set of all solutions that make the open sentence true.
To
an open sentence over a given domain, find the solution set using this domain.
Questions 1-4: Solve the open sentence over the domain 2,3,4,5 .
1.
2x  3  7
2.
8t  6t  2t
3.
n 1
is an integer
3
4.
y  y 1
A
number is any number that is positive, negative, or zero.
Subsets of Real Numbers:
Natural Numbers
Whole Numbers
Integers
It can be useful to graph the solution set of an open sentence on a number line.
5. Graph each subset of the real numbers on a number line.
a. Natural Numbers
b. Whole Numbers
c. Integers
Questions 6-7: Graph each set of numbers on a number line.
6.
3 
 5
 ,1, ,3
2 
 2
7.
the set of integers that are multiples of 4.
Questions 8-10: Solve each open sentence over the set of positive integers and graph the solution set.
8.
25  y 2  9
10.
h
is an integer
2
9.
z2  5