COLLEGE AND ADVANCED ALGEBRA
Prepared by: Mr. Vince William A. Cabotaje, LPT
Top 2, LEPT January 2022
Real Number System
Basic Operation
Addition:
Subtraction:
Multiplication:
Division:
Addend + Addend = Sum
Minuend – Subtrahend = Difference
Multiplicand × Multiplier = Product
Dividend ÷ Divisor = Quotient
Factors and Multiples
Factor- is the number than evenly divide the given number and do not leave a remainder.
Determine the factors of 12.
Multiple- is the number get after multiplying the given number by an integer.
Determine the multiples of 4.
Take note!
GCF (Greatest Common Factor)
The highest number that divides exactly into two or more numbers. It is simply the largest of the common
factors.
LCM (Least Common Multiple)
Multiple in this context is used for integers. LCM refers to the smallest number that two or more numbers
will divide without remainder.
Prime and Composites
Prime – is a number whose only factors are 1 and itself.





The largest prime number before 100 is 97.
The largest prime number before 110 is 109.
The largest prime number before 200 is 199.
There are 25 prime numbers from 1 to 100.
There are 168 prime numbers from 1 to 1000.
Prime Factorization – process of expressing a number as a product of its prime factors.
Composite – is a number that has more than two factors.
REVIEW…
Algebraic Expressions
Algebraic expressions such as
2𝑥,
8𝑥𝑦,
3𝑥𝑦2,
−4𝑎2𝑏3𝑐,
and
𝑧
are called terms. A term is an indicated product that may have any number of factors. The variables involved in a term
are called literal factors, and the numerical factor is called the numerical coefficient.
Example:
8𝑥𝑦
𝑥 and 𝑦 are literal factors
8 is the numerical coefficient
Similar Terms
Terms that have the same literal factors are called similar terms or like terms.
Example:
3𝑥 and 14𝑥
7𝑥𝑦 and −9𝑥𝑦
Simplifying Algebraic Expressions
We simplify algebraic expressions by combining similar terms.
Example:
7𝑥 + 2𝑦 + 9𝑥 + 6𝑦 = 7𝑥 + 9𝑥 + 2𝑦 + 6𝑦
= 16𝑥 + 8𝑦
More examples:
Evaluating Algebraic Expression
To evaluate an algebraic expression means to find the value of the expression when the variable is replaced
by a given number. To evaluate an expression, we substitute the given number for the variable in the
expression and then simplify the expression using the order of operations.
Evaluate:
𝑓(𝑛) = 2𝑛3 − 1 for n = 2
𝑓(2) = 2(2)3 − 1 = 2(8) − 1 = 16 − 1
𝑓(2) = 15
More examples:
Rationalizing Radical Expression
Rationalizing the denominator means to perform some operations to remove the radicals from the
denominator.
Simplify:
2
3+√3
=
=
=
=
More examples:
2
∙
3 − √3
3 + √3 3 − √3
2(3 − √3)
9 − √9
2(3 − √3)
6
3 − √3
3
=
2(3 − √3)
9−3
Linear Equation in One Variable
A linear equation in one variable (here 𝑥) is an equation that can be written in the form
𝑎𝑥 + 𝑏 = 0
where 𝑎 and 𝑏 are real numbers and 𝑎 ≠ 0.
Example:
𝑥 + 1 = −2
A linear equation is a first-degree equation because the greatest power on the variable is 1.
Solution of the Equation
If the variable in an equation can be replaced by a real number that makes the statement true, then that
number is a solution of the equation.
Example: In the equation 𝑥 − 3 = 5, we can conclude that 8 is a solution since if we replace 𝑥 with 8,
the statement will be true.
Solving a Linear Equation in One Variable
Step
Description
Example

Simplify each side
separately
Isolate the variable
terms on one side.
Isolate the variable.
Use the distributive property as
needed.
 Clear any parenthesis.
 Combine like terms.
 Use the addition property of
equality so that all terms with
variables are on one side of the
equation and all constants
(numbers) are on the other side.
Use the multiplication property of
equality to obtain an equation that has
just the variable with coefficient 1 on
one side.
2(𝑥 − 5) + 3𝑥 = 𝑥 + 6
2𝑥 − 10 + 3𝑥 = 𝑥 + 6
5𝑥 − 10 = 𝑥 + 6
5𝑥 − 10 − 𝑥 = 𝑥 + 6 − 𝑥
4𝑥 − 10 = 6
4𝑥 − 10 + 10 = 6 + 10
4𝑥 = 16
4𝑥
=
16
4
4
𝑥=4
Fractional Equations
Ratio
A ratio is the comparison of two numbers by division. We often use the fractional form to express ratios.
𝑎
For example, we can write the ratio of 𝑎 to 𝑏 as .
𝑏
Proportion
𝑎
𝑐
A statement of equality between two ratios is called a proportion. Thus, if and are two equal ratios, we
𝑎
can form the proportion = 𝑐 (𝑏 ≠ 0 and 𝑑 ≠ 0).
𝑑
𝑏
𝑏
𝑑
Cross-Multiplication Property of Proportions
𝑎
If = 𝑐 (𝑏 ≠ 0 and 𝑑 ≠ 0), then 𝑎𝑑 = 𝑏𝑐.
𝑑
𝑏
Example: Solve
5
𝑥+6
=
7
𝑥−5
Solution:
7
5
=
𝑥+6 𝑥 −5
5(𝑥 − 5) = 7(𝑥 + 6)
5𝑥 − 25 = 7𝑥 + 42
−67 = 2𝑥
−
67
2
=𝑥
Linear Inequalities in One Variable
A linear inequality in one variable (here 𝑥) is an inequality that can be written in the form
𝑎𝑥 + 𝑏 < 0, 𝑎𝑥 + 𝑏 ≤ 0,
𝑎𝑥 + 𝑏 > 0,
or
𝑎𝑥 + 𝑏 ≥ 0
where 𝑎 and 𝑏 are real numbers and 𝑎 ≠ 0.
Examples: 𝑥 + 5 < 0, 𝑥 − 3 ≥ 5
Solving a Linear Inequality in One Variable
Step
Description

Simplify each side
separately
Isolate the variable
terms on one side.
Isolate the variable.
Use the distributive property as
needed.
 Clear any parenthesis.
 Combine like terms.
 Use the addition property of
inequality so that all terms with
variables are on one side of the
equation and all constants
(numbers) are on the other side.
Use the multiplication property of
inequality in one of the following
forms, where 𝑘 is a constant (number).
variable < 𝑘, variable ≤ 𝑘,
variable > 𝑘, or variable ≥ 𝑘
*Remember: Reverse the direction of
the inequality symbol only when
multiplying or dividing each side of an
inequality by a negative number.
Example
−3(𝑥 + 4) + 2 ≥ 7 − 𝑥
−3𝑥 − 12 + 2 ≥ 7 − 𝑥
−3𝑥 − 10 ≥ 7 − 𝑥
−3𝑥 − 10 + 𝑥 ≥ 7 − 𝑥 + 𝑥
−2𝑥 − 10 ≥ 7
−2𝑥 − 10 + 10 ≥ 7 + 10
−2𝑥 ≥ 17
−2𝑥
−2
≤
17
−2
17
𝑥≤−
2
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