Transcript 2 ( x + 1 ) - Collier Youth Services

Algebra 1
Chapter 1 Notes
Introduction to Algebra
1
ALGEBRA is the process of moving values from
one side of equation to the other without changing
the equality. KEEP IT BALANCED !
If you change one side of an equation,
you must change the other side equally.
For example, if x = y, then x + 1 = y + 1
1.1
Algebraic Expressions
The Study of Algebra involves numbers and operations.
A Numerical Expression contains one of more numbers and one or more operations:
12
7.6
5+9
14 – 7 x 2
In Algebra, letters are often used to represent numbers. These letters are called
Variables.
An Algebraic Expression contains one of more variables and one or more operations:
5n
4n − 6
3y (2)
To Evaluate an Expression replace each variable with a number to find a numerical
value.
Example 1: Evaluate 5n where n = 6, thus So, 5 (6) = 30
Example 2: Evaluate 2xy for x = 4 and y = 3, thus 2 (x) (y) = 2 (4) (3) = 24
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1.2
Order of Operations
Order of Operations
1. Do all multiplications and divisions in order from left to right
2. Do all additions and subtractions in order from left to right.
Example
Simplify: a. 16 + 8 ● 9
Simplify: b. 18 − 8 ÷ 4
Solutions
a. Multiply first, then add
16 + 8 ● 9
16 + 72
88
a. Divide first, then subtract
18 − 8 ÷ 4
18 − 2
16
4
1.2
Grouping Symbols
Grouping Symbols
Parentheses ( ) and brackets [ ] are called Grouping Symbols.
The rule is to do operations within grouping symbols first.
Note: a multiplication symbol may be omitted when it occurs next
to a grouping symbol.
Example 1
3 ● (5 + 2) = 3 (5 + 2) = 3 (7) = 21
If there is more than one set of grouping symbols, operate within the
innermost symbols first.
Example 2:
5[ 8 + (7 – 3)] = 5 [8 + 4] = 5 [12] = 60
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1.3
Exponents
Exponent
The exponent indicates the number of times the vase is used as a factor.
base
53
exponent
53 = 5 ● 5 ● 5 = 125
Order of Operations
1. Operate within groups symbols first. Work from the inside to the outside.
2. Simplify powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
42 ● 13 + 8
4●4●1●1●1+8
16 ● 1 + 8 = 24
23 ● 42
(5 – 3) 2 =
2●2●2+4●4
22
8 + 16
=
4
24
= 4
= 6
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1.3
Exponents and Grouping Symbols
The exponent outside a grouping symbol differs from one where there is no
grouping symbol.
4x3 differs from (4x)3 because the exponent with a grouping symbol
raises each factor to that power. In this case (4x)3 = 43 x3 = 64x3
that any variable without an exponent is assumed to be 1.
In this example, (4x)3 , the factors inside the parentheses have an exponent of 1.
As a result, we have (41x1)3 which gives us the value as shown above
42 ● 13 + 8
4●4●1●1●1+8
16 ● 1 + 8 = 24
23 ● 42
(5 – 3) 2 =
2●2●2+4●4
22
8 + 16
=
4
24
= 4
= 6
7
Common Assumptions with Numbers
+ 1 n. 1
1
• The sign of a number is positive, +
•
The coefficient is 1
•
The decimal point is to the right of the number
•
As a whole number it is over 1
•
The power of the number is 1
1.1
Real Numbers and Number Operations
Whole numbers = 0, 1, 2, 3 …
Integers = …, -3, -2, -1, 0, 1, 2, 3 …
Rational numbers = numbers such as 3/4 , 1/3, -4/1 that can be written as a ratio of
the two integers. When written as decimals, rational numbers terminate or repeat,
3/4 = 0.75, 1/3 = 0.333…
Irrational numbers = real numbers that are NOT rational, such as, 2 and  , When
written as decimals, irrational numbers neither terminate or repeat.
Origin

-3
-2
-1
0
1
2
3
A Graph of a number is a point on a number line that corresponds to a real number
The number that corresponds to a point on a number line is the Coordinate of the
point.
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1.1
Real Numbers and Number Operations
Graph - 4/3, 2.7,
2
•
-5 -4 -3
-1
• •
0
1
2
•
Graph - 2, 3
-5 -4 -3
Graph - 1, - 3
-2
4
5
•
-2
-1
-2
-1
•
-5 -4 -3
3
0
1
2
3
4
5
0
1
2
3
4
5
•
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1.1
Real Numbers and Order of Operation
Example: You can use a number line to graph and order real numbers.
Increasing order (left to right): - 4, - 1, 0.3, 2.7
-5 -4 -3
-2
-1
0
1
2
3
4
5
Properties of real numbers include the closure, commutative, associative, identity, inverse
and distributive properties.
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1.1
Using Properties of Real Numbers
Properties of addition and multiplication [let a, b, c = real numbers]
Property
Addition
Multiplication
Closure
a + b is a real number
a • b is a real number
Commutative
a+b=b+a
a•b=b• a
Associative
(a+b)+c=a+(b+c)
(ab)c=a(bc)
Identity
a+0=a, 0+a=a
a•1=a,1•a=a
Inverse
a + ( -a ) = 0
a • 1/a = 1 , a  0
Distributive
a ( b + c) = a b + a c
Opposite = additive inverse, for example a and - a
Reciprocal = multiplicative inverse (of any non-zero #) for example a and 1/a
Definition of subtraction: a – b = a + ( - b )
Definition of division: a / b = a 1 / b , b  0
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1.1
Real Numbers and Number Operations
Identifying properties of real numbers & number operations
(3+9)+8=3+(9+8)
14 • 1 = 14
[ Associative property of addition ]
[Identity property of multiplication ]
Operations with real numbers:
Difference of 7 and – 10 ?
7 – ( - 10 ) = 7 + 10 = 17
•
- 10
•
- 5
-4 -3
-2
-1
0
1
2
3
4
5
7
Quotient of - 24 and 1/3 ?
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1.1
Real Numbers and Number Operations
Give the answer with the appropriate unit of measure
A.) 345 miles – 187 miles = 158 miles
“Per” means divided by
B.) ( 1.5 hours ) ( 50 miles ) = 75 miles
1 hour
C) 24 dollars = 8 dollars per hour
3 hours
D) ( 88 feet ) ( 3600 seconds ) ( 1 mile ) = 60 miles per hour
1 second
1 hour
5280 feet
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1.1
Solve Linear Equations
Identifying Properties
33. – 8 + 8 = 0
34. ( 3 • 5 ) • 10 = 3 • ( 5 • 10 )
35. 7 • 9 = 9 • 7
36.
(9+2)+4=9+(2+4)
37. 12 (1) = 12
38. 2 ( 5 + 11 ) = 2 • 5 + 2 • 11
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1.1
Solve Word Problems
Operations
43. What is the sum of 32 and – 7 ?
44. What is the sum of – 9 and – 6 ?
45. What is the difference of – 5 and 8 ?
46. What is the difference of – 1 and – 10 ?
47. What is the product of 9 and – 4 ?
48. What is the product of – 7 and – 3 ?
49. What is the quotient of – 5 and – ½ ?
50. What is the quotient of – 14 and 7/4 ?
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1.1
Solve Unit Measures
Unit Analysis
51. 8 1/6 feet + 4 5/6 feet =
52. 27 ½ liters – 18 5/8 liters =
53. 8.75 yards ( $ 70 ) =
1 yard
54. ( 50 feet ) ( 1 mile ) ( 3600 seconds ) =
1 second
5280 feet
1 hour
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1.2
Algebraic Expressions and Models
Order of Operations
1.First, do operations that occur within grouping symbols
2.Next, evaluate powers
3.Do multiplications and divisions from left to right
4.Do additions and subtractions from left to right
Numerical expression:
2
- 4 + 2 ( -2 + 5 ) = - 4 + 2 (3 )
=-4+2(9)
= - 4 + 18
= 14
2
5
2 =2• 2 •2 •2 •2
[ 5 factors of 2 ] or [ 2 multiplied out 5 times ]
In this expression:
the number 2 is the base
the number 5 is the exponent
the expression is a power.
A variable is a letter used to represent one or more numbers. Any number used to replace
variable is a value of the variable. An expression involving variables is called an algebraic
expression. The value of the expression is the result when you evaluate the expression by
replacing the variables with numbers.
An expression that represents a real-life situation is a mathematical model. See page 12.
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1.2
Algebraic Expressions and Models
Example: You can use order of operations to evaluate expressions.
2
8 (3 + 4 ) – 12  2 =
8 (3 + 16) – 6 =
8 (19) – 6 =
152 – 6 = 146
Numerical expressions:
Algebraic expression:
2
3 x – 1 when x = – 5
2
3 (– 5 ) – 1 =
3 (25) – 1 = 74
Sometimes you can use the distributive property to simplify an expression.
Combine like terms:
2
2 x – 4 x + 10 x – 1 =
2
2 x + (– 4 + 10 ) x – 1 =
2
2x +6x-1
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1.2
Evaluating Powers
4
Example 1: ( - 3 ) = ( - 3 ) ( - 3 ) ( - 3 ) ( - 3 ) = 81
4
- 3 = - ( 3 3 3 3 ) = - 81
Example 2: Evaluating an algebraic expression
2
- 3 x – 5 x + 7 when x = - 2
2
- 3 ( - 2 ) – 5 ( - 2 )x + 7 [ substitute – 2 for x ]
2
- 3 ( 4 ) – 5 ( - 2 )x + 7
[ evaluate the power, 2 ]
- 12 + 10 + 7
[ multiply ]
+5
[ add ]
Example 3: Simplifying by combining like terms
a)
7 x + 4 x = ( 7 + 4 ) x [ distributive ]
= 11 x
[ add coefficients ]
b)
3 n + n – n = ( 3 n – n ) + n [ group like terms ]
2
= 2n +n
[ combine like terms ]
c)
2 ( x + 1 ) – 3 ( x – 4 ) = 2 x + 2 – 3 x + 12
[ distributive ]
= ( 2 x – 3 x ) + ( 2 + 12 ) [ group like terms ]
= - x + 14
[ combine like terms ]
2
2
2
2
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1.3
Solving Linear Equations
Transformations that produce equivalent equations
Additional property of equality
Add same number to both sides
if a = b, then a + c = b + c
Subtraction property of equality
Subtract same number to both sides
if a = b, then a - c = b - c
Multiplication property of equality
Multiply both sides by the same number
if a = b and c ǂ 0, then a • c = b • c
Division property of equality
Divide both sides by the same number
if a = b and c ǂ 0, then a ÷ c = b ÷ c
Linear Equations in one variable in form a x = b, where a & b are constants and a ǂ 0.
A number is a solution of an equation if the expression is true when the number is substituted.
Two equations are equivalent if they have the same solution.
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1.3
Solve Linear Equations
Solving for variable on one side [by isolating the variable on one side of equation ]
Example 1: 3 x + 9 = 15
7
3 x + 9 - 9 = 15 - 9
7
3x=6
7
[ subtract 9 from both sides to eliminate the other term ]
7• 3x= 7•6
3 7
[ multiply both sides by 7/3, the reciprocal of 3/7, to get x by itself]
x = 14
Example 2: 5 n + 11 = 7 n – 9
-5n
-5n
[ subtract 5 n from both sides to get the variable on one side ]
11 = 2 n – 9
+9
+ 9 [ add 9 to both sides to get rid of the other term with the variable ]
20 = 2 n
2
2
[ divide both sides by 2 to get the variable n by itself on one side ]
10 = n
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1.3
Solve Linear Equations
Example: You can use properties of real numbers and transformations that produce
equivalent equations to solve linear equations.
Solve
4 ( 3 x – 5 ) = – 2 (– x + 8 ) – 6 x
Write original equation
12 x – 20 = 2 x – 16 – 6 x
Use distributive property
12 x – 20 = – 4 x – 16
Combine Like Terms
16 x – 20 = – 16
Add 4 x to both sides
16 x = 4
x = 1/4
Add 20 to both sides
Divide each side by 16
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1.3
Solve Linear Equations
Equations with fractions
Example 3: 1 x + 1 = x – 1
3
4
6
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1.4
ReWriting Equations and Formulas
Example: You can an equation that has more than one variable, such as a formula, for one
of its variables.
Solve the equation for y:
2x–3y= 6
–3y= –2x+6
y = 2x–2
3
Solve for the formula for the area of a trapezoid for h:
A = 1 ( b1 + b2) h
2
2A =
2A =
( b1 + b2)
( b1 + b2) h
h
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1.4
ReWriting an Equation with more than 1 variable
Solve : 7 x – 3 y = 8 for the variable y.
7x–3y= 8
-7x
- 7 x [ subtract 7 x from both sides to get rid of the other term ]
–3 y= 8 – 7 x
–3
–3 –3
[divide both sides by – 3 to get the variable x by itself on one side ]
y = –8 + 7 x
3
3
Calculating the value of a variable
Solve: x + x y = 1 when x = – 1 and x = 3
x + x y = 1 [ first solve for y so that when you replace x with – 1 and 3, you also solve for y ]
-x
- x [ subtract x from both sides to get rid of the other term without y in it ]
xy=1–x
x
x [divide by x to get y by itself ]
y = 1–x
x
when x = - 1, then y = - 2 and when x = 3, then y = - 2/3
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1.4
Common Formulas
Distance
D=rt
d = distance, r = rate, t = time
Simple interest
I=prt
I = interest, p = principal, r = rate, t = time
Temperature
F = 9/5 C + 32
F = degrees Fahrenheit, C = degrees Celsius
Area of a Triangle
A=½bh
A = area, b = base, h = height
Area of a Rectangle
A=lw
A = area, l = length, w = width
Perimeter of Rectangle
P=2l+2w
P = perimeter, l = length, w = width
Area of Trapezoid
A = ½ ( b1 + b2 ) h
A = area, b1 = 1 base, b2 = 2 base, h = height
Area of Circle
A=
Circumference of Circle
C=2πr
2
πr
A = area, r = radius
C = circumference, r = radius
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