A.
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.
Don’t be afraid !
This will not hurt !
A.
What Is Algebra
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.
A.
What Is Algebra
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. 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”
1. INTEGERS
Every place on the line represents an integer.
Integer values include:
• All the positive whole numbers
• Zero
• All the negative whole numbers
Integer Examples
0
-5
1. INTEGERS
Negative integers located to left of zero.
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.
1. INTEGERS
Positive integers located to right 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.
1. INTEGERS
Plus sign (+) is used to:
• indicate the addition operation.
• indicate positive integer value.
Minus sign (-) is used to:
• indicate the subtraction operation.
• indicate negative integer value.
Zero has no sign.
2. COMPARING THE VALUE OF INTEGERS
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).
Larger digit, but less value because of position on line.
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”.
The expression is always read from left to right, like a sentence.
REMEMBER . .
When working with “negative” integers, a number’s value is
determined by its place on the number line, not by the value of
the whole number digit.
3. ABSOLUTE VALUES
Definition:
The absolute value of an integer is its value without regard
to the sign.
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.
Understanding Integers Exercise
greater than > 22
1. 27 is ________________
greater than > -6
2. 10 is ________________
less than <
3. -37 is ________________
40
4. Determine the absolute values.
a. |-20| = _________
20
25
b. |25| = _________
c. |-1|
= _________
1
d. |-129| = _________
129
e. |0| = ________
0
Check your answers !
4. ADDING SIGNED INTEGERS
Exact procedure depends on whether addends have same
signs 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.
4. ADDING SIGNED INTEGERS
ADDING INTEGERS THAT HAVE THE SAME SIGN
Step 1: Add the absolute values of the addends.
Step 2: Give the result the sign that is common.
(+ 6) or 6
Adding two positive integers: (+2) + (+4) = __________
(- 6)
Adding two negative integers: (-2) + (- 4) = __________
4. ADDING SIGNED INTEGERS
ADDING INTEGERS THAT HAVE OPPOSITE SIGNS
Step 1: Subtract the absolute values of the addends.
Step 2: Write the sum with the sign of the larger
number.
Adding a positive integer to a negative integer:
(+2) + (-4) = - 2
Adding a negative integer to a positive integer:
(-2) + (+4) = + 2 or just 2
5. SUBTRACTING SIGNED INTEGERS
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 Addition
The process is always simpler when we change the operation
from subtraction to addition
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)
5. SUBTRACTING SIGNED INTEGERS
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
6. COMBINING INTEGER ADDITION AND SUBTRACTION
Some equations require adding or subtracting three or more
signed integers.
Examples:
(+4) + (+5) + (+12) + (+6) = ?
(+17) – (+24) + (-1) – (+6) = ?
• Always perform the operations from left to right.
(+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
6. COMBINING INTEGER ADDITION AND SUBTRACTION
Equations with mixed-sign integers:
(+12) + (-14) + (-8) = ?
• Add the terms, two at a time, from left to right.
(+12) + (-14) + (-8) =
(-2) + (-8) = (-10)
What happens when the equation includes both addition and subtraction?
6. COMBINING INTEGER ADDITION AND SUBTRACTION
Equations with mixed signs and mixed operations:
(+17) - (+24) + (-1) – (+6) = ?
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
6. COMBINING INTEGER ADDITION AND SUBTRACTION
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
1.
(+ 6) + ( 5) + (4) + ( 8) = (23)
2.
(+ 8) – (+ 1) + (+ 4) + (– 2) +( – 5) = (4)
3.
( + 7) – ( + 5) + (– 1) = (+1)
4.
(+ 7) + (+ 4) – (+ 6) = (+5)
5.
(+ 6) – (+ 1) + (– 2) – (– 9) = (+12)
1.
(– 2) + (– 1) – (– 6) = (+3)
2.
( 6) – (+ 7) – (+ 1) + (+ 3) = (+1)
3.
(+ 3) – (+ 7) – (+ 8) – (+ 2) – (+ 3) = (-17)
Check your answers !
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.
Step 2: Give the appropriate sign to the product.
 Positive if both factors have the same sign (even if both are –).
 Negative if the factor have opposite signs.
NOTE: Zero has no sign.
7. MULTIPLYING SIGNED INTEGERS
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
7. MULTIPLYING SIGNED INTEGERS
MULTIPLYING INTEGERS HAVING OPPOSITE SIGNS
If signs are opposite – one (+), one (-) – product is always negative.
Step 1: Multiply the two factors – disregard the sign.
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
7. MULTIPLYING SIGNED INTEGERS
LESSON SUMMARY
To multiply integers that have the same sign (both + or both -):
Step 1: Multiply the two factors – disregard the sign.
Step 2: Show product as positive integer.
To multiply integers that have opposite signs:
Step 1: Multiply the two factors – disregard the sign.
Step 2: Show product as negative integer.
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|
Example 2:
(+32)
(-8) = |+32|
|-8| = + 3 (same signs – pos. quotient)
|-8| = - 4 (opposite signs – neg. quotient)
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)
(+24)
(-8) = (+24)
(-8)
(-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 Exercise
1. 2 x 4 x (-1) = - 8
2. - 4 x 2 x (-6) = - 48
3. 8 . 6 . 4 . -2 = - 384
4. (4) (5) (-2) (2) = -80
5. 2 x 6
3= +4
6. 2 x 4
(-1) = - 8
7. 64
8. 4
(16) x (6) = 24
(2) (12)
-8 = - 3
Check your answers !
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
10. INTRODUCING EXPONENTS
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
“One raised to any power equals 1”
or
09 = 0
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
10. INTRODUCING EXPONENTS
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
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+8
0+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
4x8=
24
4x8=
6 x 8 = 48
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”.
When solving combinations of addition, subtraction, multiplication,
and division in the same expression:
Order of Precedence
1.
•
- Examples
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
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 Exercises
Cite the value of these “powered” integers.
1. 22 = 4
2. 33 = 27
3. 52 = 25
4. 82 = 64
5. 34 = 81
Simplify these equations, using the correct order of precedence.
6. 8 + 16
4 – 6 + 2 x 3 = 8 + (16
7. 24 – 12
6 + 4 + 2 x 3 = 24 - (12
6. 2 x 4 + 18 – 33
3=
4) – 6 + (2 x 3) = 12
6) + 4 + (2 x 3) = 32
(2 x 4) + 18 – (33
3) = 15
Check your answers !
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
32 = 3 x 3 = 9
Base
Example 2:
Notation
n1 = n
45 = 4 x 4 x 4 x 4 x 4 = 1024
Explanation
Any number with an exponent
of 1 is equal to that number itself.
Example
51 = 5
n0 = 1
Any number with an exponent
of 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
The opposite of squaring a number is taking the square root.
•
•
The square of 4 is 16.
The square root of 16 is 4.
Radical Sign
4
16
Radicand
Don’t worry about the square root of negative numbers !
Squares and Square Roots for Integers 1 - 9
Squares
12
22
32
42
52
62
72
82
92
102
=1
=4
=9
= 16
= 25
= 36
= 49
= 64
= 81
= 100
Square Roots
1=1
4=2
9=3
16 = 4
25 = 5
36 = 6
49 = 7
64 = 8
81 = 9
100 = 10
14. INTRODUCING EXPRESSIONS AND EQUATIONS
Replacing 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
An 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.
Expressions
X
3 + 5 = 4y - 7
Equation
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
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
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 + 18
15 + 18 = 33
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
a – b = c just means “variable minus variable = variable
X–2=8
Don’t be confused by use of “x”.
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 Exercises
c. 3 to the fifth power
1. The number 35 means: ______________________
2. In exercise 1 above:
base
a. The number 3 is called the _________
exponent
b. The number 5 is called the _________
3. True or False:
a. ____
T The number 9 is the square of 3.
F The number 9 is the square root of 18.
b. ____
c. ____
T The number 9 is the square root of 81.
d. ____
T The number 90 = 1
4. The difference between an algebraic expression and an algebraic equation is:
b. The algebraic equation contains an equal sign
5. Combine the like terms in the following expressions:
7a - 8
a. 4a + a + 2a – 8 = ___________________.
4x + 2
b. 6x - 2x + 3 – 1 = ___________________.
10x – 4y
c. 3x + 7x – 4y = _____________________.
10x – 4y
d. 12 + 2 + 5y – 2x = __________________.
Check your answers !
Expression 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
Y – 5 + 5 = 13 + 5
Y = 18
8. 2x + 11 = 43
2x + 11 - 11 = 43 - 11
2x = 32
X = 16
9. 4a – 9 = 15
4a – 9 + 9 = 15 + 9
4a = 24
a=6
10. x(3 + 4) – 8 = 20
3x + 4x -8 = 20
7x – 8 = 20
7x – 8 + 8 = 20 + 8
7x = 28
x=4
Check your answers !