Finite Math and Elementary Calculus
Section 1.1 Jigsaw
REAL NUMBERS, INEQUALITIES, AND LINES

Group 1
REAL NUMBERS AND INEQUALITIES
1.
What is a real number?
A number that can be represented by a point on the number line (also called the
real line).
2.
Where do we typically graph real numbers? Give an example.
On a number line (the real line)
3.
What is used to express the order of the real numbers? __inequalities____
Use words to express the following inequalities:
a ≤ b
a
b
a < b ___a
b________________________________________ a ≥ b ___a
b____________________________ a > b ___a
b______________________________________
4.
What does the double inequality a < x < b mean? both the inequalities a
 x and

hold_____
5.
When graphing inequalities on a number line, remember that a solid dot  indicates that the point is included in the interval and a hollow dot indicates that the point is
excluded. Graph a < x < b
on the number line below.
SET NOTATION
6.
Braces { } are read as _”the set of all” and a vertical bar | is read __”such that”__.

EXAMPLES
1.
Insert the appropriate symbol from above into the following inequality.
1
100
________ -1,000,000
2.
Use words to describe the intervals expressed in set notation below. a.
{ x | x > 3 } ___the set of all x such that x is greater than 3____________________________ b.
{ x | -2 < x < 5} _”the set of all x such that x is greater than -2 AND less than 5” (or “x is between
–2 and 5”)._____________
3.
Write the following in set notation:
“The set of all x such that x is greater than or equal to -7.”
4.
Graph { x
|
2 x 5
} on the number line below.

Group 2
INTERVAL NOTATION
1.
Interval notation is another way of expressing or writing down a _________________
______________________________________ .
2.
In interval notation, square brackets [ and/or ] indicate that ___________________
_______________________________________ .
3.
In interval notation, parentheses indicate that _______________________________
__________________________________ .
4.
An interval is ________________ if it includes both endpoints, and ____________ if it
includes neither endpoint.
5.
Show the interval notation and number line graphs for the following sets. Tell whether the interval is open or closed.
Set Notation Interval
Notation
Graph Open or
Closed
b
x
6.
When an interval extends infinitely far to the right or to the left, it is said to be an
_____________________________ interval.
7.
Show the interval notation and number line graph for an infinite interval.
Interval Notation: _____________________________
Graph:

8.
The symbol denotes ______________________________________________.
EXAMPLES
9.
Write the following set of real numbers in set notation, then graph on the real line below.
Interval Notation: (-2, 6) Set Notation: ______________________________
Graph:
10.
Write the following set in interval notation, then graph on the real line below.
Set Notation:
 x a x b

Interval Notation: _____________________________
Graph:
11.
Write the following interval in set notation and graph it on the real line below.
[ 4, )
Set Notation: _______________________________
Graph:

Group 3
CARTESIAN PLANE
1.
Two real lines or axes, one horizontal and one vertical, intersecting at their ___________
points, define the _______________________________.
2.
On a Cartesian plane, the two axes divide the plane into four _______________________.
3.
Any point in the Cartesian plane can be specified uniquely by an _________________________
_____________ of numbers ______________.
4.
In an ordered pair of numbers
( , )
, the x-coordinate is the number on the
________________ axis corresponding to the point, and the y-coordinate is the number on the _________________ axis corresponding to the point.
5.
Label the four quadrants in the Cartesian plane below.
6.
The symbol  (read “delta,” the Greek letter D) means ______________________________.
7.
For any two points x y
1 1
)
and
( ,
2 2
)
, we define x
__________ and y
_____________.
8.
Any two distinct points determine a _______________.
9.
A nonvertical line has a _____________ that measures the steepness of the line defined as
________________________________ divided by ________________________________

for any two points on the line.
10.
The formula for the slope of a line through x y
1 1
)
and m
 _____________ = ________________ x y
2 2
)
is:
11.
The changes  x
and  y
are often called, respectively, the “___________” and the
“__________” with the understanding that a negative “rise” means a “fall.”
EXAMPLES
1.
Find the slope (if it is defined) of the line determined by each pair of points, then graph the line. a.
(2, 3)
and
(4, 1) m
 ____________ b.

1
2
and
(5, )
2 m
 _____________ c.
(6, 4)
and  m
 _____________

Group 4
EQUATIONS OF LINES
1.
The point where a nonvertical line crosses the y-axis is called the ______________________ of the line and can be given either as the _____________________________ or as the point __________ .
2.
A nonvertical line can be expressed very simply in terms of its ______________ and
________________________, representing the points on the line by variable coordinates (or
“variables”) x
and y .
3.
Fill in the table below showing the forms of equations of lines with which you are familiar.
Form of Equation of Line
Slope-Intercept Form
Equation Definitions of Variables and
Constants Used m
 slope; b
 y
 intercept
Point-Slope Form m
 slope; x y
1 1
)
 point on line
General Form
Vertical Line
Horizontal Line a , b ,
and fractions or decimals) a
is the x
 intercept b
is the y c
must be integers (no
 intercept
EXAMPLES
1.
Find an equation of the line with slope -2 and y-intercept 4, and graph it.

2.
Find both the point-slope and slope-intercept forms of the equation of the line through (6, -2) with slope -
1
.
2
3.
Find the general form equation for the line through the points (4, 1) and (7, -2).
4.
Find an equation for the vertical line through the point (5, 7).
5.
Find an equation for the horizontal line through the point (5, 7).
6.
Find the slope and y-intercept of the line 2x + 3y = 12.