**Outline**

Today's lesson will cover some basic concepts of arithmetic.

Toward the end of the lesson, we will solve a few problems using

these concepts. The chapters that will be covered day are:

1.3 and
1.5-Elementary Arithmetic

1.8, 1.9,
and 1.10-Applications

**1.3 Arithmetic**

The **commutative** law says that when you add or multiply

numbers, the order you add or multiply them does not matter. In

other words,

a + b = b + a.

The ** distributive ** law says that when a number is multiplied by the

sum of two numbers that it equals, that number multiplied by the

two numbers individually, with the results added afterward, that is,

(a + b) · c = a ·c + b · c.

**1.3 Order of Operations**

When a number is both preceded and followed by two binary

operations that are not the same, a rule is required for which

operation should be applied first, this rule is known as the **order of**

operations. So binary operations within parentheses outrank

exponents, exponents outrank multiplication and division (but

multiplication and division have equal rank), and these two

outrank addition and subtraction (which have equal rank).

**1.5 Adding and Subtracting Fractions**

When adding fractions with the same denominator, the result is

the same as the sum of the numerators over the denominator. A

similar statement can be made for subtraction. If the fractions

have different denominators ,

First, find the least common denominator.

Then write equivalent fractions using this denominator.

Add or subtract the fractions.

**1.5 Adding and Subtracting Mixed Numbers**

To add or subtract mixed numbers, simply convert the mixed

numbers into improper fractions, then add or subtract them as

fractions.

**Example :**

**1. 5 Multiplying Fractions and Fractions**

When two fractions are multiplied, the result is a fraction with a

numerator that is the product of the fractions ' numerators and a

denominator that is the product of the fractions' denominators.

Example :

**1.5 Multiplying Mixed Numbers**

To multiply mixed numbers, convert them to improper fractions

and multiply.

**Example**:

**1.5 Reciprocal**

The reciprocal of a fraction is obtained by switching its numerator

and denominator. To find the reciprocal of a mixed number, first

convert the mixed number to an improper fraction, then switch the

numerator and denominator of the improper fraction. Notice that

when you multiply a fraction and its reciprocal, the product is

always 1.

**Example:**

Find the reciprocal of . We switch the
numerator and

denominator to find the reciprocal:.

**1.5 Dividing Fractions**

To divide a number by a fraction, multiply the number by the

reciprocal of the fraction.

**Example:**

**1.5 Dividing Mixed Numbers**

To divide mixed numbers, you should always convert to improper

fractions, then multiply the first number by the reciprocal of the

second.

**Example:**

**1.8 Percentage**

To convert a fraction to a percentage, divide the numerator by the

denominator. Then move the decimal point two places to the right

(which is the same as multiplying by 100) and add a percent sign.

**Example:** Given the fraction what is the
percentage?

To change a percentage to a fraction, divide it by 100 and reduce

the fraction or move the decimal point to the right until you have

only integers.

**1.9 Proportions**

A proportion is a name we give to a statement that two ratios are

equal. It can be written in two ways:

or

**1.13 Calculations: converting from one unit to another**

The simplest way to carry out calculations involving different units

is to use the dimensional method. In this method, a quantity

described in one unit is converted to another by means of a

conversion factor. A conversion factor is ratio of two quantities

that have different units:

start quantity × conversion factor = equivalent quantity:

**1.15 Integers**

The set containing the natural numbers, zero, and all negative

numbers is called the integers. The symbol Z is used to denote set

of integers. Some important properties of the integers:

The sum or product of two integers is always an integer.

The integers is an ordered set.

**1.5 Addition and Subtraction of Integers**

Subtracting of an integer is equivalent to adding the opposite

or inverse of that integer.

The **absolute value** of an integer represents the distance of

that integer from zero.

When adding two integers of the same, add the absolute value

of the integers. The sign of the sum is the same as sign of

both integers.

When adding two integers of opposite signs, de termine their

absolute value and then subtract the smaller number from

larger number. The sign of the difference is the sign of larger

number

**1.16 Multiplication and Division of Integers**

The product or quotient of two integers of the same sign is

always positive.

The product or quotient to two integers of opposite sign is

always negative.

The quotient of two integers is not always an integer.

**1.16 Multiplication and Division of Integers**

The product or quotient of two integers of the same sign is

always positive.

The product or quotient to two integers of opposite sign is

always negative.

The quotient of two integers is not always an integer.

**4.1 Inequality symbols **

≠ means not equal to

< means is less than

> means is greater than

≤
means less than or equal to

≥
means greater than or equal to

**4.2 Solving inequalities**

For real numbers a, b, and c.

If a < b, then a + c < b + c.

If a < b, then a + c < b - c.

Similar statements can be made for the symbols >, ≤, and ≥.

For real numbers a, b, and c.

If a < b and c > 0, then ac < bc.

If a < b and c < 0, then ac > bc.

If a < b and c > 0, then a/c < b/c.

If a < b and c < 0, then a/c > b/c.

Similar statements can be made for the symbols >, ≤, and ≥.

**4.3 Exponents and Polynomials **

If n is a natural number, then

If n and m are natural numbers, then

If n and m are natural numbers, then

If n is a natural number, then and if y ≠
0,

then

**4.4 The FOIL method**

FOIL is an acronym for **F**irst terms, **O**uter terms,** I**nner terms, and

**L**ast terms. To use the FOIL method to multiply 2x - 4 by 3x + 5,

we

Multiply the **F**irst terms 2x and 3x to get 6x^{2},

Multiply the **O**uter terms 2x and 5 to get 10x,

Multiply the** I**nner terms -4 and 3x to get -12x, and

Multiply the** L**ast terms -4 and +5 to get -20.

**4.5 Factoring Trinomials of the form x**^{2} + bx + c

Find two integers whose product is constant term in the

trinomial.

There sum must equal the coefficient of the middle term.

**4.6 Factoring Trinomials of the form ax**^{2} + bx + c

Find the factors of the leading coefficient.

Find the factors of the last term.

Try combination of the first terms and last terms until you

find one that works.

**3.4 Linear functions **

The slope of a line function is defined by

Where

**
3.4 Geometric Meaning of Slope**

From its geometric meaning, slope may be determined from a

graph by the formula

**
3.5 Horizontal and Vertical Intercepts**

**Definition: **A horizontal intercept of a graph is a point where the

graph crosses the horizontal axis. The ordered pair notation for a

horizontal intercept has the form (a, 0), where a is the input value.

**Definition:** A vertical intercept of a graph is a point where the

graph crosses the vertical axis. The ordered pair notation for a

vertical intercept has the form (0, b), where b is the output value.