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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUES AND INEQUALITY SYMBOLS

 

newton Sir Isaac Newton((1642-1727)

Sir Isaac Newton was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is considered by many scholars and members of the general public to be one of the most influential people in human history. His 1687 publication of the Philosophiæ Naturalis Principia Mathematica (usually called the Principia) is considered to be among the most influential books in the history of science, laying the groundwork for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the scientific revolution.

SECTION 4.1: SIGNED NUMBERS

A "signed number" is a digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) or combination of digits with a positive(+) or negative(-) sign in from of it.

A signed number with "+" symbol in front of it is called a "positive number". In general, mathematicians do not place a "+" symbol in front of a number to designate a positive number. If there is no "sign symbol" in front of a number, the number is "understood" to be positive.

A signed number with "-" symbol in front of it is called a "negative number".

ex. 1) +7 is called positive 7, but most books will write 7 for +7 

ex. 2) -12 is called negative 12

 

 

SECTION 4.2: OPPOSITE NUMBERS

The numbers 3 and -3, 5 and -5, 2/3 and -2/3, etc. are called "opposite numbers". Opposite numbers are made up of the same digits in the same order but have opposite signs.

 

SECTION 4.3: ABSOLUTE VALUES

The following symbolism, | 7 |, is asking you a question. It's asking, "How many units is 7 from 0 on the number line?" Your answer would be 7 units from 0.

7

| -5 | is asking, "How many units is -5 from 0 on the number line?" The answer is 5 units from 0.

neg5

The symbol, |  |, is called the "absolute value symbol". The number inside the absolute value symbol is called the "argument," and the result you get is called the "absolute value".

MEANING OF THE ABSOLUTE VALUE

The absolute value represents a distance from 0.
Distance is a positive number by agreement.

ex. 1) | -6 | = 6; and 6 is called the absolute value and -6 is 6 units from 0

ex. 2) | 6 | = 6; and 6 is called the absolute value and 6 is 6 units from 0

The distance from 0 to 6 and 0 to -6 is exactly the same. The distance is 6 units.

Figure

ex. 3) | -25 | = 25; and 25 is called the absolute value and -25 is 25 units from 0

 

SECTION 4.4: ADDING POSITIVE NUMBERS

At this point in your math career you know you can add positive numbers, add positive and negative numbers, and add negative numbers. We'll review each one of these cases.

RULE FOR ADDING POSITIVE NUMBERS

Use traditional addition. The result is understood to be positive.


ex. 1) 5 + 10 = +15             ex. 2) 3/5 + 7/5 = +10/5               ex. 3) 2/3 + 3/4 = +17/12

Comment: In general the positive symbol "+" is dropped in an answer. So +15 would be written as 15.

 

SECTION 4.5: ADDING NEGATIVE NUMBERS

 

RULE FOR ADDING NEGATIVE NUMBERS

Step 1) Find the absolute value of each number

Step 2) Add the absolute values

Step 3) The result of step 2 takes a negative sign.

 

ex. 1) -3 + -5

Step 1) | -3 | = 3 and | -5 | = 5

Step 2) 3 + 5 = 8

Step 3) ans: -8

 

ex. 2) -7 + -11

Step 1) | -7 | = 7 and | -11 | = 11

Step 2) 7 + 11 = 18

Step 3) ans: -18




SECTION 4.6: ADDING A POSITIVE AND A NEGATIVE NUMBER

RULE FOR ADDING A POSITIVE AND A NEGATIVE NUMBER

Step 1) Find the absolute value of each number

Step 2) Subtract the smallest absolute value from the largest absolute value.

Step 3) The result in step 2 is given the sign of the argument with the largest absolute value.

 

ex. 1) -11 + 7

step 1) | -11 | = 11 and | 7 | = 7

step 2) 11 - 7 = 4

step 3) Of 11 and 7 in step 1, 11 is larger.
Take the sign of the argument of | -11 |.
The argument is -11: the sign is negative.
The negative sign is given to the result in step 2.

ans: -4

 

ex. 2) -6 + 24

step 1) | -6 | = 6 and | 24 | = 24

step 2) 24 - 6 = 18

step 3) Of 6 and 24 in step 1 , 24 is larger.
Take the sign of the argument of the | 24 |.
The argument is +24: the sign is positive.
The positive sign is given to the result in step 2.

ans: +18 or 18



SECTION 4.7: SUBTRACTING SIGNED NUMBERS

 

RULE FOR SUBTRACTING SIGNED NUMBERS

Please note that a "signed number" is a number with
with either a positive or negative sign.

Step 1) Change the sign of the number to the right of subtraction to the opposite sign.

Step 2) Change the subtraction sign to addition.

Step 3) Now apply the rule of adding signed numbers.

Teacher Comment: I know this rule may seem weird but think about 7 - 2.

7 - 2 can be rewritten as 7 + (-2)

 

ex. 1) -12 - -11

step 1) -12 - 11

I changed the sign of the number to the right of subtraction, -11, to the opposite sign to get 11.

step 2) -12 + 11

I changed the subtraction sign to addition.

step 3) I will use the rules of adding signed numbers to determine -12 + 11.

-12 + 11 = -1.

Therefore, -12 - -11 = -1

ex. 2) 17 - 21

step 1) 17 - -21

I changed the sign of the number to the right of subtraction, 21, to the opposite sign to get -21.

step 2) 17 + -21

I changed the subtraction sign to addition.

step 3) I will use the rules of adding signed numbers to determine 17 + -21.

17 + -21= -4.

Therefore, 17 - 21 = -4

 

--- AN ASIDE ---
DEFINITIONS OF SUM AND DIFFERENCE

The "sum of two numbers x and y" is the result of adding the numbers x and y.

ex. ) 5 + 7 = 12 and 12 is the sum

The "difference of two numbers x and y" is the result of subtracting the number y from x.

ex. ) 12 - 7 = 5 and 5 is the difference

 



SECTION 4.8: RULES FOR MULTIPLYING AND DIVIDING SIGNED NUMBERS

 

What is a "product"? It is the result of multiplying numbers. So, in 4 x 7 = 28, 28 is the product.

 

MULTIPLICATION RULE FOR MULTIPLYING TWO SIGNED NUMBERS

When multiplying 2 SIGNED numbers:

Step 1) Multiply the absolute values of the numbers

Step 2) If the two ORIGINAL numbers have signs that are

a) positive: the product in step 1) takes a positive sign.

b) negative: the product in step 1) takes a positive sign.

c) opposites: the product in step 1) takes a negative sign

ex. 1) 5 x 7
s1) |5| x |7|= 35
s2a) + 35
ex. 2) -5 x -7
s1) |-5| x |-7|= 35
s2b) + 35
ex. 3) 5 x -7
s1) |5| x |-7|= 35
s2c) - 35

 

What is a "quotient"? It is the result of dividing numbers. So, in 40/5 = 8, 8 is the quotient.

 

DIVISION RULE FOR DIVIDING TWO SIGNED NUMBERS

When dividing 2 SIGNED numbers:

Step 1) Divide the absolute values of the numbers

Step 2) If the two ORIGINAL numbers have signs that are

a) positive: the quotient in step 1) takes a positive sign.

b) negative: the quotient in step 1) takes a positive sign.

c) opposite: the quotient in step 1) takes a negative sign

 

ex. 1) 5/7   s1) |5| / |7| = 5/7  s2a) + 5/7
ex. 2) -5/-7   s1) |-5| / |-7| = 5/7 s2b) + 5/7
ex. 3) 5/-7  s1) |5| / |-7| = 5/7 s2c) - 5/7

 


SECTION 4.9: INEQUALITY SYMBOLS

Instead of using the word phrases "less than," "greater than," "not equal to," "less than or equal to" or "greater than or equal to," mathematicians have some symbols to take the place of these word phrases. These symbols are called "inequality symbols."

The inequality symbols are as follows:

Figure

Figure

 

 

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THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES