1.2 – Open Sentences and Graphs

Definitions

An expression is a number, a variable, or a sum, difference, product, or quotient that contains one or more variables.

Examples: 214,275,3

4,97

y

x

Definitions

A variable is a symbol, usually a letter, that represents any of the members of a specified set. This set is called the domain of the variable, and its members are called the values of the variable.

Definitions

A mathematical sentence is a group of symbols that states a relationship between two mathematical expressions. These can be either true or false.

Examples: 6814,1721,5

2

4

3

Definitions

An open sentence 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: 93,52,47 hmpmx

Definitions

The values of the variable that make an open sentence true are called the solutions of the open sentence.

The solution set is the set of all solutions that make the open sentence true.

To solve an open sentence over a given domain, find the solution set using this domain.

Questions 1-4: Solve the open sentence over

the domain

1. 2.

3. is an integer 4.

5,4,3,2

732 x ttt 268

3

1n 1yy

Definitions

A real number is any number that is positive, negative, or zero.

Subsets of Real Numbers: Natural Numbers:

Whole Numbers:

Integers:

,...3,2,1

,...3,2,1,0

,...3,2,1,0,1,2,3...,

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

Graph each set of numbers on the number line.

6.

7. The set of integers that are multiples of 4.

3,

2

3,1,

2

5

Solve each open sentence over the set of positive integers and graph the solution set.

8.

9.

10. Is an integer

925 2 y

52 z

2

h