a. what is algebra a branch of mathematics in which arithmetic relations are generalized and...

A. What Is AlgebraA. What Is Algebra

A branch of mathematics in which arithmetic relations are generalized and explored using letter symbols to represent numbers, variable quantities, and other mathematical entities.

This will not hurt !

Don’t be afraid !


Named in honor of Islamic mathematician Mohammed ibn Musa al-Khowarizmi:

• 825 AD

• Wrote book “Hisb al-jabr wa’l muqubalah” (The Science of Reduction and Cancellation)

• Book “Al-jabr” presented rules for solving equations.

Algebra:

• is a Problem Solving tool.

• applies to every human endeavor.

• is a tool in art to medicine to zoology.


Algebra takes work, but you can learn it.

• Probably already done algebra in elementary school.

• Remember problems like 5 + ? = 8 (really an algebraic equation)?

• Algebra uses letter like “x” instead of “?”

To avoid confusion:

• “x” is not used to indicate multiplication.

• sometimes we use a raised dot “.”

• sometimes we write nothing between letters.

Example: “r” times “s” may be written:

r.s rs (r)(s) r(s) (r)s

1. INTEGERS1. INTEGERS

• In Basic Math, we use Arabic numerals (0,1,2,3,4,5,6,7,8,9)

• Those numbers make up all whole numbers and fractions.

Algebra explores different uses of digits

• necessary to call them something else – “INTEGERS”.

Integers are the set of “all” whole numbers, both positive and negative, including zero.

INTEGER LINE

“Infinity” symbol means “continue as far as required”

Integer values include:

• All the positive whole numbers

• Zero

• All the negative whole numbers

Every place on the line represents an integer.

Integer Examples

01

- 93627

-5

-2748

53


Negative value integers:

• Use same figures as whole number system

• Distinguished by use of negative sign (-).

• Numbers 5 and -5 appear similar but are entirely different values.

Negative integers located to left of zero.


Positive value integers:

• Sometimes indicated by positive sign (+).

• Most often, positive sign is omitted.

• Integer values with no sign are assumed to be positive.

Positive integers located to right of zero.


Minus sign (-) is used to:

• indicate the subtraction operation.

• indicate negative integer value.

Plus sign (+) is used to:• indicate the addition operation.• indicate positive integer value.

Zero has no sign.


Values of integers increase from left to right on number line.

Values of integers decrease from right to left on number line.

Increasing values

Decreasing values

4 > 1 “means 4 is greater than one” (to right of one on number line).

0 < 3 “means 0 is less than three” (to left of three on number line).

-1 < 4 “means -1 is less than four” (to left of +4 on number line).

-3 > -5 “means -3 is greater than -5” (to right of -5 on number line).

2. COMPARING THE VALUE OF INTEGERS 2. COMPARING THE VALUE OF INTEGERS

< means “less than”. (always points to smaller “less than” value).

> means “greater than”. (always points to larger “greater than” value).

4 < 5 “means 4 is less than 5”.

3 > 2 “means 3 is greater than 2”.

-8 < -4 “means -8 is less than -4”.

REMEMBER . .

When working with “negative” integers, a number’s value isdetermined by its place on the number line, not by the value ofthe whole number digit.

The expression is always read from left to right, like a sentence.

2. COMPARING THE VALUE OF INTEGERS 2. COMPARING THE VALUE OF INTEGERS

Definition:

The absolute value of an integer is its value without regard to the sign.

or, put another way . . .or, put another way . . .

The absolute value of an integer is its distance from the origin (zero) on the number line.

Absolute value is indicated by enclosing numbers in a Pair of vertical lines | |.

Example: The absolute value of -10 is written as |-10| .

Value is 10.

3. ABSOLUTE VALUES 3. ABSOLUTE VALUES

Understanding Integers ExerciseUnderstanding Integers Exercise

1. 27 is ________________ 22

2. 10 is ________________ -6

3. -37 is ________________ 40

4. Determine the absolute values.a. |-20| = _________

b. |25| = _________

c. |-1| = _________

d. |-129| = _________

e. |0| = ________

Check your answers !

greater than >

greater than >

less than <

20

25

1

129

0

4. ADDING SIGNED INTEGERS4. ADDING SIGNED INTEGERS

Exact procedure depends on whether addends have samesigns or opposite sign.

Same Signs (both + or both -) Add the absolute values of the addends. Give the result the sign that is common.

Opposite Signs (one + and one -) Subtract the absolute values of the addends. Give the result the sign of the addend that has the larger absolute value.

ADDING INTEGERS THAT HAVE THE SAME SIGN

Adding two positive integers: (+2) + (+4) = __________

Adding two negative integers: (-2) + (- 4) = __________

Step 1: Add the absolute values of the addends.

Step 2: Give the result the sign that is common.

(+ 6) or 6

(- 6)


ADDING INTEGERS THAT HAVE OPPOSITE SIGNS

Adding a positive integer to a negative integer:

Adding a negative integer to a positive integer:

Step 1: Subtract the absolute values of the addends.

Step 2: Write the sum with the sign of the larger number.

(+2) + (-4) = - 2

(-2) + (+4) = + 2 or just 2


Step 1: Change the subtraction sign to the addition sign, then switch the sign of the subtrahend (the number the immediately follows the operation sign you just changed).

Step 2: Add the result according to the procedures for adding signed integers.

Changing Integer Subtraction to Integer AdditionThe process is always simpler when we change the operation from subtraction to addition

5. SUBTRACTING SIGNED INTEGERS5. SUBTRACTING SIGNED INTEGERS

As you study further into algebra you will notice that practically all of the equations are formed around the equal (=) sign.

Equal (=) means “Everything on the left side of the sign has the same value as everything on the right side. Both are “equal in value”.

Example:

(+2) – (+6) = (+2) + (-6)

COMPLETING THE OPERATION

Example:(+15) – (+12)

• Change from subtraction to addition.• Switch the sign of the number that immediately follows.

(+15) – (+12) = (+15) + (-12)

Add the result.

(+15) + (-12) = (+3) or just 3

5. SUBTRACTING SIGNED INTEGERS5. SUBTRACTING SIGNED INTEGERS

6. COMBINING INTEGER ADDITION AND SUBTRACTION 6. COMBINING INTEGER ADDITION AND SUBTRACTION

Examples:(+4) + (+5) + (+12) + (+6) = ?

• Always perform the operations from left to right.

Some equations require adding or subtracting three or more signed integers.

(+17) – (+24) + (-1) – (+6) = ?

(+4) + (+5) + (+12) + (+6) = ?

• Add the terms, two at a time, from left to right.

(+4) + (+5) + (+12) + (+6) = (+9) + (+12) + (+6)

(+9) + (+12) + (+6) = (+21) + (+6)

(+21) + (+6) = (+27)

Just like a regular addition problem: 4 + 5 + 12 + 6 = 27

(+12) + (-14) + (-8) = ?

Equations with mixed-sign integers:

(+12) + (-14) + (-8) =

• Add the terms, two at a time, from left to right.

(-2) + (-8) = (-10)

What happens when the equation includes both addition and subtraction?


(+17) - (+24) + (-1) – (+6) = ?

Equations with mixed signs and mixed operations:

Step 1: Change the subtraction signs (-) to addition (+).

Step 2: Switch the sign attached to the term that follows the operation.

(+17) - (+24) + (-1) – (+6) =

(+17) + (-24) + (-1) + (-6) =

(-7) + (-1) + (-6) =

(-8) + (-6) = (-14)

Simplified: 17 – 24 – 1 – 6 = -14


Simplifying Signed-Integer Expressions For Addition and Subtraction:

Parentheses are often overused.

(+2) is the same as 2

(–2) is the same as –2

(+2) + (+3) is the same as 2 + 3

(+2) – (+3) is the same as 2 – 3

(–2) – (+3) is the same as –2 – 3

(+2) – (–3) is the same as 2 + 3


ADDING AND SUBTRACTING SIGNED INTEGERS EXERCISES

ADDING AND SUBTRACTING SIGNED INTEGERS EXERCISES

1. (+ 6) + ( 5) + (4) + ( 8) =

2. (+ 8) – (+ 1) + (+ 4) + (– 2) +( – 5) =

3. ( + 7) – ( + 5) + (– 1) =

4. (+ 7) + (+ 4) – (+ 6) =

5. (+ 6) – (+ 1) + (– 2) – (– 9) =

1. (– 2) + (– 1) – (– 6) =

2. ( 6) – (+ 7) – (+ 1) + (+ 3) =

3. (+ 3) – (+ 7) – (+ 8) – (+ 2) – (+ 3) =

(23)

(4)

(+1)

(+5)

(+12)

(+3)

(+1)

(-17)


7. MULTIPLYING SIGNED INTEGERS 7. MULTIPLYING SIGNED INTEGERS

Basic procedure identical to multiplying whole numbers. The only difference is dealing with the + and – signs.

Step 1: Multiply the absolute value of the factors.

Positive if both factors have the same sign (even if both are –).

Negative if the factor have opposite signs.

NOTE: Zero has no sign.

Step 2: Give the appropriate sign to the product.


MULTIPLYING INTEGERS HAVING THE SAME SIGN

If signs are same – positive or negative – product is always positive.

Step 1: Multiply the two factors – disregard the sign.

Step 2: Show product as positive integer.

Example 1:

(+5) x (+2) =

Multiply absolute value of terms:|+5| x |+2| = 10

Assign appropriate sign to product:+10

Example 2:(-8) x (-3) = 8 x 3 = 24


MULTIPLYING INTEGERS HAVING OPPOSITE SIGNS

If signs are opposite – one (+), one (-) – product is always negative.


Step 2: Show product as negative integer.

Example :

(-7) x (+2) =

Multiply absolute value of terms:|-7| x |+2| = 14

Assign appropriate sign to product:-14


LESSON SUMMARY

To multiply integers that have the same sign (both + or both -):


Step 2: Show product as positive integer.

To multiply integers that have opposite signs:


Step 2: Show product as negative integer.

8. DIVIDING SIGNED INTEGERS 8. DIVIDING SIGNED INTEGERS

Process is basically identical to procedure for multiplying:

• Divide the absolute value of the terms.

• Give the appropriate sign to the quotient. Positive if terms both have the same sign. Negative if the terms have opposite signs.

Example 1:(-24) (-8) = |-24| |-8| = + 3 (same signs – pos. quotient)

Example 2:(+32) (-8) = |+32| |-8| = - 4 (opposite signs – neg. quotient)

9. COMBINING INTEGER MULTIPLICATION 9. COMBINING INTEGER MULTIPLICATION

Always perform the operations from left to right.

Example 1:(+12) x (+2) (-8) =

Multiply or divide the terms two at a time, from left to right.

(+12) x (+2) (-8) = (+24) (-8)

(+24) (-8) = (-3)

Example 2:(+16) (-8) x (-6) =

Multiply or divide the terms two at a time, from left to right.

(+16) (-8) x (-6) = (-2) x (-6)

(-2) x (-6) = (+12)

Multiplying and Dividing Signed Integers ExerciseMultiplying and Dividing Signed Integers Exercise

1. 2 x 4 x (-1) =

2. - 4 x 2 x (-6) =

3. 8 . 6 . 4 . -2 =

4. (4) (5) (-2) (2) =

5. 2 x 6 3 =

6. 2 x 4 (-1) =

7. 64 (16) x (6) =

8. 4 (2) (12) -8 =


- 8

-80

- 8

- 3

24

+ 4

- 48

- 384

10. INTRODUCING EXPONENTS 10. INTRODUCING EXPONENTS

Recall: Multiplication is a short-cut method for adding groups of equal numbers.

3 + 3 + 3 + 3 = 4 x 3 = 12

Short-cut method uses “exponential notation”.

2 x 2 x 2 x 2 = 16 Expressed in exponential notation is 24 = 16

Spoken as “two raised to the fourth power”.

Powers of 2, or “Squares”

• 2 is most common exponent

• Numbers raised to power of two are said to be “squared”.

Examples:

“Three squared equals nine” 32 = 9

“Five squared equals twenty five” 52 = 25

“Ten squared equals one hundred” 102 = 100


Powers of 3, or “Cubes”

• 3 is another common exponent

• Numbers raised to power of three are said to be “cubed”.

Examples:

“Two cubed equals eight” 23 = 8 (2 x 2 x 2 = 8)

“Three cubed equals twenty seven” 33 = 27 (3 x 3 x 3 = 27)

“Ten cubed equals one thousand” 102 = 100 (10 x 10 x 10 = 1000)

Special Cases:

“Zero raised to any power equals zero” 02 = 0 or 09 = 0

“One raised to any power equals 1” 13 =1

“Any value raised to the 0 power equals 1” 20 =1“Confusing, but that’s the rule”.

“Any value raised to the 1 power equals itself” 51 = 5


Exponents of Signed Integers

• The sign of any squared value is always positive.

Examples:“The square of any number is always positive.

• The square of a positive number is a positive value. 32 = 3 x 3 = 9

• The square of a negative number is a positive value. (- 4)2 = ( - 4) (- 4) = 16

11. ORDERING OPERATIONS WITH INTEGERS 11. ORDERING OPERATIONS WITH INTEGERS

Basic Rules of Algebra

• When solving combinations of addition and subtraction operations on three or more terms, do the operations from left to right.

Example:2 + 5 – 7 + 8 =7 – 7 + 8 = 0 + 80 + 8 = 8

• When solving combinations of multiplication and division operations on three or more terms, do the operations from left to right.

Example:2 x 12 4 x 8 = 24 4 x 8 =6 x 8 = 48

11. ORDERING OPERATIONS WITH INTEGERS 11. ORDERING OPERATIONS WITH INTEGERS

Order of Precedence

• What do you do when solving combinations of: - addition ,subtraction, multiplication and division ? - expression enclosed in parentheses ? - terms with exponents ?

The specific rules for these operations are called “order of operation or order of precedence”.

Order of Precedence - ExamplesOrder of Precedence - Examples

1. Simplify 4 + 2 x 6 = ?• Multiply first: 4 + 2 x 6 = 4 + 12

• Add last: 4 + 12 = 16

• The solution is: 4 + (2 x 6) = 16 THE SOLUTION IS NOT (4 + 2) X 6 = 36

2. Simplify 6 + 18 6 = ?• Divide first: 6 + 18 6 = 6 + 3

• Add last: 6 + 3 = 9

• The solution is: 6 + (18 6) = 9 THE SOLUTION IS NOT (6 + 18) 6 = 4

3. Simplify 4 + 3 x 6 – 4 + 8 x 2 = ?• Multiply first, from left to right: 4 + (3 x 6) – 4 + (8 x 2) = 4 + 18 – 4 + 16

• Add/ subtract last, from left to right: (4 + 18) – 4 + 16 = 22 – 4 + 16

(22 – 4) + 16 = 18 + 16

18 + 16 = 34

• The solution is: 4 + 3 x 6 – 4 + 8 x 2 = 34

Order of Precedence Order of Precedence

The order of operation for combination problems is:1st = Parentheses

2nd = Exponents

3rd = Multiplication, Division (left-to-right)

4th = Addition, Subtraction (left-to-right)

15 – 9 x 23 + (24 6) – 11 = ?

1st = Parentheses

2nd = Exponents

3rd = Multiplication, Division (left-to-right)

4th = Addition, Subtraction (left-to-right)

Exponents & Order of Operation Practice ExercisesExponents & Order of Operation Practice Exercises

Cite the value of these “powered” integers.

1. 22 =

2. 33 =

3. 52 =

4. 82 =

5. 34 =

Simplify these equations, using the correct order of precedence.

1. 8 + 16 4 – 6 + 2 x 3 =

2. 24 – 12 6 + 4 + 2 x 3 =

1. 2 x 4 + 18 – 33 3 =


4

27

25

64

81

8 + (16 4) – 6 + (2 x 3) = 12

24 - (12 6) + 4 + (2 x 3) = 32

(2 x 4) + 18 – (33 3) = 15

12. INTRODUCING POWER NOTATION 12. INTRODUCING POWER NOTATION

Power Notation – A method for indicating the power of a number.

• The base indicates the number to be multiplied.• The exponent indicates the number of times the base is to be multiplied.

Example 1: 32 Exponent

Base

32 = 3 x 3 = 9

45 = 4 x 4 x 4 x 4 x 4 = 1024Example 2:

Notation Explanation Example

n1 = n Any number with an exponentof 1 is equal to that number itself.

51 = 5

n0 = 1 Any number with an exponentof 0 is equal to 1.

30 = 1

1k = 1 1 to any power is equal to 1. 14 = 1

0k = 0 0 to any power is equal to 0. 05 = 0

n-k = 1 nk

Any number with a negative exponent is equal to 1 divided by that number with a positive exponent.

2-3 = 1 = 0.125 23

13. INTRODUCING SQUARE ROOTS 13. INTRODUCING SQUARE ROOTS

The opposite of squaring a number is taking the square root.

• The square of 4 is 16.• The square root of 16 is 4.

Don’t worry about the square root of negative numbers !

Squares and Square Roots for Integers 1 - 9

Squares Square Roots

12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81102 = 100

1 = 1 4 = 2 9 = 3 16 = 4 25 = 5 36 = 6 49 = 7 64 = 8 81 = 9100 = 10

Radical SignRadicand

416

14. INTRODUCING EXPRESSIONS AND EQUATIONS 14. INTRODUCING EXPRESSIONS AND EQUATIONSReplacing numbers with letters.

• 2 is always 2• 6 is always 6• letters can mean anything, for example:

Compare: Arithmetic Expression: 2 + 1 Algebraic Expression: x + 1

(can only be 3)(depends on the value of “x”)

A lot more flexible.

Definitions

• A specific numerical value (such as 2,4, -6, ¾) is called a constant.

• An algebraic term (such as x, y, a, b) is called a variable.

• When constants and/or variables are connected by operations (such as +, -, x, . ), you have an expression.

14. INTRODUCING EXPRESSIONS AND EQUATIONS 14. INTRODUCING EXPRESSIONS AND EQUATIONSAn equation is a statement of equality between two expressions.

It consists of two sets of algebraic expressions separated by an equal sign.

Purpose is to express equality between the two expressions.

An equation includes an equal sign (=), an expression does not.

An expression can include signs of operation, but not an equal sign.

X 3 + 5 = 4y - 7

Expressions

Equation

15. EVALUATUNG ALGEBRAIC EXPRESSIONS 15. EVALUATUNG ALGEBRAIC EXPRESSIONS

An algebraic equation can be evaluated by:• Assigning specific numerical values to all of the variables.

• Completing all the operations.

Example 1

Evaluate the expression x + 9 ( when x = 5)

• Replace the given value of x in the expression:

X + 9 = 5 + 9

• Complete the operation:

5 + 9 = 14

• Solution:

X + 9 = 14 (when x = 5)

Review example 2 in manual.

16. COMBINING LIKE TERMS 16. COMBINING LIKE TERMS

Like terms are expressions that have the same variable:• 2x and 4x

• y and 5y

Example 1 – Combine Like Terms

1. 2x + 4x = 6x

2. 3x + 2x + 4 = 5x + 4

3. y + 3y + 2x = 4y + 2x

Notice that you can only combine the like terms, the ones with the same variables.

You can’t combine 3 apples + 4 apples + 6 oranges and get 13 apples –you get 7 apples and 6 oranges.

17. USING THE DISTRIBUTIVE PROPERTY 17. USING THE DISTRIBUTIVE PROPERTY

Deals with the combinations of multiplication and addition,or multiplication and subtraction.

Allows you to remove parentheses and simplify equations.

Standard rule is for addition is: a(b + c) = ab + ac

Standard rule is for subtraction is: a(b - c) = ab - ac

Multiplier is distributed across both variables.

Evaluate a( b+c ), where a = 3, b = 5, c = 6

Substitute values: 3 (5 + 6)

Do the operations: (multiplication first) 3*5 + 3*6 = 15 + 1815 + 18 = 33

18. SOLVING EQUATIONS 18. SOLVING EQUATIONS

An equation is a mathematical statement of equality between two expressions.

Solving equations of the form a + b = c and a – b = c

Example of form: a – b = c

X – 2 = 8

Don’t be confused by use of “x”.

a – b = c just means “variable minus variable = variable

Strategy is to make “x” stand alone on left side of equal sign.

Remember: Whatever is done to one side of the equation must also be done to the other side.

X – 2 + 2 = 8 + 2

X + 0 = 10

X = 10

Expression and Equation ExercisesExpression and Equation Exercises

1. The number 35 means: ______________________

2. In exercise 1 above: a. The number 3 is called the _________ b. The number 5 is called the _________

3. True or False: a. ____ The number 9 is the square of 3. b. ____ The number 9 is the square root of 18. c. ____ The number 9 is the square root of 81. d. ____ The number 90 = 1

4. The difference between an algebraic expression and an algebraic equation is:

5. Combine the like terms in the following expressions: a. 4a + a + 2a – 8 = ___________________. b. 6x - 2x + 3 – 1 = ___________________. c. 3x + 7x – 4y = _____________________. d. 12 + 2 + 5y – 2x = __________________.


c. 3 to the fifth power

baseexponent

T

TF

T

b. The algebraic equation contains an equal sign

7a - 84x + 210x – 4y

10x – 4y

Expression and Equation ExercisesExpression and Equation Exercises

6. Use the distributive property to evaluate: a(b+c), where a=2, b=4, c=7


2(4 + 7) = 8 + 14 = 22

7. Y – 5 = 13

8. 2x + 11 = 43

9. 4a – 9 = 15

10. x(3 + 4) – 8 = 20

Y – 5 + 5 = 13 + 5Y = 18

2x + 11 - 11 = 43 - 112x = 32X = 16

4a – 9 + 9 = 15 + 94a = 24a = 6

3x + 4x -8 = 207x – 8 = 207x – 8 + 8 = 20 + 87x = 28 x = 4

a. what is algebra a branch of mathematics in which arithmetic relations are generalized and...

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value of integers

positive integers

negative integers

positive value integers

values of integers

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numbers value