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1. PRECALCULUS 1
(ALGEBRA AND TRIGONOMETRY)
CHAPTER 1: REAL NUMBER AND THEIR PROPERTIES,
SYSTEM OF LINEAR EQUATIONS, GRAPHS AND THEIR
APPLICATION
PREPARED BY:
ENGR. RAYMOND JAY G. SEVERO

2. INTRODUCTION:
Algebra uses letters, such as x and y, to represent numbers.
If a letter is used to represent various numbers, it is called a
variable. A combination of variables and numbers using the
operations of addition, subtraction, multiplication, or division,
well as powers or roots, is called an algebraic expression. Many
Many algebraic expressions involve exponents.
Example:
4x2 + 341x + 3194 = 0

3. Exponential Notation:
If “n” is a counting number (1, 2, 3,
and so on), on the figure. bn is read
“the nth power of b ” or “ b to
the nth power.” Thus, the nth
power of b is defined as the product
of n factors of b. The expression
bn is called an exponential expression.

4. SETS
A set is a collection of objects whose contents can be clearly
determined. The objects in a set are called the elements of the
set.
Roster method sample, the set of numbers used for counting can be
represented by {1, 2, 3, 4, 5 …}. The braces, { }, indicate that we are
representing a set.
Set-builder notation in this notation, the elements of the set are
described but not listed.

5. The intersection of sets A and B, written
A ∩ B, is the set of elements common to
both set A and set B.
The union of sets A and B, written A ∪ B, is the
set of elements that are members of set
A or of set B or of both sets. This definition can
be expressed in set-builder notation as follows:
A ∪ B = {x l x is an element of A or x is an
element of B}.
Solution: {7, 8, 9, 10, 11} and {6, 8, 10, 12}
are 8 and 10.
Answer: {7, 8, 9, 10, 11} ∩ {6, 8, 10, 12} = {8,
10}.
Example: Find the union:
{7, 8, 9, 10, 11} ∪ {6, 8, 10, 12}.
Solution: Just rewrite all the numbers encase the
two sets have same numbers, just write it once {7, 8,
9, 10, 11} ∪ {6, 8, 10, 12},
Answer: {7, 8, 9, 10, 11} ∪ {6, 8, 10, 12} = {6, 7, 8,
9, 10, 11, 12}.

6. Application Problems (Venn Diagram)
1. In a class of 40 students, 27 like Calculus and 25 like Chemistry.
How many like both Calculus and Chemistry?
2. A club of 40 executives, 33 like to smoke Marlboro, and 20 likes
smoke Philip Morris. How many like both?
3. A survey of 100person s revealed that 72 of them had eaten at
restaurant P and that 52 of them had eaten at restaurant Q.
of the following could not be the number of persons in the
surveyed group who had eaten at both P and Q restaurant?

7. 4. In a certain party each one of the group drinks coke or beer or
whisky or all. Also 400 drinks coke, 500 drink beer and 300 drink
whisky. 100 drinks coke and beer and 200 drinks beer and whisky. One
who drinks whisky does not drink coke. How many are in the group?
5. A survey of 100 students reported that the number of those enrolled
in various Mathematics subjects were, Algebra, Geometry and Calculus
20 enrollees; Algebra and Geometry 30 enrollees; Algebra and Calculus
35 enrollees; Geometry and Calculus 35 enrollees, Algebra only 70
enrollees and Calculus only 60 enrollees. How many enrolled in
Geometry?

8. 6. The President just recently appointed 25 Generals of the
Philippine Army. Of these 14 have already served in the war of
Korea, 12 in the war of Vietnam and 10 in the war of Japan. There
are 4 who serve both Korea and Japan, 6 have serve both Vietnam
and Korea, and 3 have serve both in Japan and Vietnam. How many
have served in Japan, Korea and Vietnam?

10. Exercises No. 1: Consider the following set of numbers: {-7, -
3/4, 0, 0.6, 81, 𝜋, 7.3, 8, }.
List the numbers in the set that are:
a. Natural numbers ______________________
b. Whole numbers ______________________
c. Integers ______________________
d. Rational numbers ______________________
e. Irrational numbers

11. REAL NUMBERS AND THEIR PROPERTIES
Real Numbers are used in everyday life to describe quantities
such as age, miles per gallon, and population. Real numbers are
represented by symbols such as -2, 5, 0, 4/3 , 0.667 . . . , 12.6, 12, π and
1238.
Ordering Real Numbers
One important property of real numbers is
that they are ordered.
The symbols <, >, ≤, and ≥ are inequality
symbols.

12. Exercises No.2:
Identify the subset of number represented in the figures.

13. Exercises No.2:
Sketch the subset on the real number line and state
whether the interval is bounded or unbounded.
1. -2 < x < 2
2. 0 ≤ x ≤ 4
3. [-2, 3)

14. Absolute value and distance
The absolute value of a real number is its magnitude, or the distance
between the origin and the point representing the real number on the real
number line.

15. Exercises No.3:
1. Evaluate
𝑥
𝑥
for
a. x > 0
b. x < 0
2. Evaluate 𝑥𝑦
a. x = 3 and y = -5
b. x = -4 and y = -12
3. Evaluate
𝑥
𝑦
a. x = 15 and y = -3
b. x = -24 and y = -4
4. Determine the distance of two
points from the given figure.

16. Algebraic Expressions
Here are a few examples of algebraic expressions.
2x, 3x + 5, 2x2 - 4x + 16, 6x2+4x+4, 5x + 7y

17. Basic Rules of Algebra
There are four arithmetic operations with real numbers:
addition, multiplication, subtraction, and division, denoted by
the symbols +, x or *, –, and ÷ or /. Of these, addition and
multiplication are the two primary operations. Subtraction
division are the inverse operations of addition and
multiplication, respectively.

22. RECTANGULAR COORDINATES
The Cartesian Plane
Just as you can represent real numbers by points on a real number line,
you can represent ordered pairs of real numbers by points in a plane called
the rectangular coordinate system, or the Cartesian plane, named after the
the French mathematician René Descartes (1596–1650).

23. The Pythagorean Theorem and the Distance Formula
Pythagorean Theorem
For a right triangle with hypotenuse of length and sides of lengths a and b
you have as a2 + b2 = c2. (The converse is also true. That is, if a2 + b2 = c2, then
the triangle is a right triangle.)
Distance Formula
The distance d between two points (x1, y1) and (x2,
y2) in the plane
d = (𝑥2 – 𝑥1)2 + (𝑦2 − 𝑦1)2

24. Midpoint Formula
The midpoint of the line segment joining the points (x1, y1) and (x2,
y2) is given by the Midpoint Formula:
Midpoint = (
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
)
Definition of Function
A function f from a set A to a set B is
a relation that assigns to each element
x in the set A exactly one element y in
the set B. The set A is the domain (or
set of inputs) of the function f, and the
set B contains the range (or set of
outputs).

25. Four Ways to Represent a Function
1. Verbally by a sentence that describes how the input variable is
related to the output variable
2. Numerically by a table or a list of ordered pairs that matches
input values with output values
3. Graphically by points on a graph in a coordinate plane in which
the input values are represented by the horizontal axis and the
output values are represented by the vertical axis.
4. Algebraically by an equation in two variables

26. Linear Equation in Two Variables
Using Slope
The simplest mathematical model for relating two variables is the
linear equation in two variables y = mx + b. The equation is called
linear because its graph is a line.
y = mx +b
where m is slope and b is y-intercept

27. Finding the Slope of a Line
Given an equation of a line, you can find its
slope by writing the equation in slope-intercept
form. If you are not given an equation, you can
still find the slope of a line. For instance,
suppose you want to find the slope of the line
passing through the points (x1, y1) and (x2, y2) as
shown in Figure. As you move from left to right
along this line, a change of (y2 – y1) units in the
vertical direction corresponds to a change of (x2
– x1) units in the horizontal direction.
Slope = rise/ run
Slope = y2 – y1 /x2 – x1

28. The Slope of a Line Passing Through Two points
The slope m of the non-vertical line through (x1, y1) and (x2, y2) is
m =
𝒚 𝟐−𝒚 𝟏
𝒙 𝟐−𝒙 𝟏
, where x1 ≠
x2
Point-Slope Form of the Equation of a Line
The equation of the line with slope m passing through the point (x1, y1) is
y – y1 = m (x – x1)
Parallel and Perpendicular Lines
1. Two distinct non-vertical lines are parallel if and only if their slopes are equal.
That is, m1 = m2.
2. Two non-vertical lines are perpendicular if and only if their slopes are
negative reciprocals of each other. That is, m1 = -1/m2.

29. Year, t Amount,
AA
1990 475
1991 577
1992 521
1993 569
1994 609
1995 562
1996 707
1997 723
1998 718
1999 648
2000 495
2001 476
2002 527
2003 464
Year, t Value V
0 12,000
1 10,750
2 9,500
3 8,250
4 7,000
5 5,750
6 4,500
7 3,250
8 2,000
Exercises No.4:
1. Plot the following function and connect the points by a line.

30. Exercises No.4:
2. Find the midpoint of the line segment joining the points (-5, -3) and
(9,3)
3. Find the midpoint of the line segment joining the points (-2, -6) and
(5,10)
4. FedEx Corporation had annual revenues of $20.6 billion in 2002 and
$34.7 billion in 2006. Without knowing any additional information,
would you estimate the 2004 revenue to have been? (Source: FedEx
Corp.)

31. Additional Problems:
1. A line has a parametric equation of x = 4 + 3t and y = 7 + t. Find the
intercept of the line.
2. A line has an equation of x + 5y + 5 = 0. (a) Find the equation of the
through point (3, 1) that is perpendicular to this line. (b) Find the
equation of the line through (3, 1) that is parallel to this line.
3. (a) Find the value of y if a line having a slope of 4/3 and passing
point (6, 4) intersects the y-axis. (b) What is the length of the line from
point (6, 4) to the y-axis. (c) Find the value of x if this line intersects the
x-axis from point (6, 4).
4. The slope of a line passing through the centroid is equal to 4/3. One
point is at (1, 2). (a) If the centroid of the line is located at (4, y) find
value of y. (b) What is the distance of the point to the centroid? (c)
Determine the equation of the line.

32. End of Chapter 1
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