Algebra 1Algebra 1

Chapter 1 NotesChapter 1 Notes

Introduction to Introduction to AlgebraAlgebra

1

ALGEBRAALGEBRA is the process of moving values from is the process of moving values from one side of equation to the other without changing one side of equation to the other without changing

the equality. the equality. KEEP IT BALANCED !KEEP IT BALANCED !

If you change one side of an equation, If you change one side of an equation, you must change the other side you must change the other side

equally.equally. For example, if x = y, then x + 1 = y + 1 For example, if x = y, then x + 1 = y + 1

Algebraic ExpressionsAlgebraic 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

1.11.1

3

Order of OperationsOrder 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.

1.21.2

4

Example Simplify: a. 16 + 8 ● 9 Simplify: b. 18 − 8 ÷÷ 4

Solutions a. Multiply first, then add16 + 8 ● 916 + 72

88

a. Divide first, then subtract18 − 8 ÷÷ 4

18 − 216

Grouping SymbolsGrouping 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.

1.21.2

5

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

ExponentsExponents

Exponent

The exponent indicates the number of times the vase is used as a factor.

1.31.3

6

53baseexponent

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.

53 = 5 ● 5 ● 5 = 125

42 ● 13 + 84 ● 4 ● 1 ● 1 ● 1 + 816 ● 1 + 8 = 24

23 ● 42 =

2 ● 2 ● 2 + 4 ● 4 =

8 + 16 =

24= 6(5 – 3) 2 22 4 4

Exponents and Grouping SymbolsExponents and Grouping Symbols

The exponent outside a grouping symbol differs from one where there is no grouping symbol.

1.31.3

7

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

42 ● 13 + 84 ● 4 ● 1 ● 1 ● 1 + 816 ● 1 + 8 = 24

23 ● 42 =

2 ● 2 ● 2 + 4 ● 4 =

8 + 16 =

24= 6(5 – 3) 2 22 4 4

Common Assumptions with NumbersCommon Assumptions with Numbers

+ 1 n. + 1 n. 11

11

• The sign of a number is positive, The sign of a number is positive, ++• The coefficient is The coefficient is 11• The decimal point is to the right of the numberThe decimal point is to the right of the number• As a whole number it is over As a whole number it is over 11• The power of the number is The power of the number is 11

2

Origin

-3 -2 -1 0 1 2 3

Real Numbers and Number OperationsReal 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, and , When written as decimals, irrational numbers neither terminate or repeat.

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.

1.11.1

9

Graph - 4/3, 2.7, 2

Graph - 2, 3

Graph - 1, - 3

• •

• •

••

•

1.11.1Real Numbers and Number OperationsReal Numbers and Number Operations

10

Example: You can use a number line to graph and order real numbers.

Increasing order (left to right): - 4, - 1, 0.3, 2.7- 4, - 1, 0.3, 2.7

Properties of real numbers include the closure, commutative, associative, identity, inverse and distributive properties.

11

Real Numbers and Order of OperationReal Numbers and Order of Operation 1.11.1

Using Properties of Real NumbersUsing Properties of Real Numbers 1.11.1

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 ) ( a b ) c = a ( b c )

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 a and 1/a

Definition of subtraction: a – b = a + ( - b )

Definition of division: a / b = a 1 / b , b 012

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

••

Quotient of - 24 and 1/3 ?

Real Numbers and Number OperationsReal Numbers and Number Operations 1.11.1

13

Real Numbers and Number OperationsReal Numbers and Number Operations 1.11.1

Give the answer with the appropriate unit of measure

A.) 345 miles – 187 miles = 158 miles

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

“Per” means divided by

14

Solve Linear EquationsSolve Linear Equations 1.11.1

15

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

Solve Word ProblemsSolve Word Problems 1.11.1

16

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 ?

Solve Unit MeasuresSolve Unit Measures 1.11.1

17

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

Algebraic Expressions and ModelsAlgebraic Expressions and Models 1.21.2

18

Order of Operations1.First, do operations that occur within grouping symbols - 4 + 2 ( -2 + 5 ) 2 = - 4 + 2 (3 ) 2

2.Next, evaluate powers = - 4 + 2 ( 9 )3.Do multiplications and divisions from left to right = - 4 + 184.Do additions and subtractions from left to right = 14

Numerical expression: 25 = 2 • 2 • 2 • 2 • 2[ 5 factors of 2 ] or [ 2 multiplied out 5 times ]

In this expression:the number 2 is the basethe number 5 is the exponentthe 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.

Algebraic Expressions and ModelsAlgebraic Expressions and Models 1.21.2

19

Example: You can use order of operations to evaluate expressions.

Numerical expressions: 8 (3 + 48 (3 + 422) – 12 ) – 12 2 = 2 =

8 (3 + 16) – 6 = 8 (3 + 16) – 6 = 8 (19) – 6 =8 (19) – 6 = 152 – 6 = 146152 – 6 = 146

Algebraic expression: 3 x3 x22 – 1 when x = – 5 – 1 when x = – 5

3 (– 5 )3 (– 5 )22 – 1 = – 1 =

3 (25) – 1 = 743 (25) – 1 = 74

Sometimes you can use the distributive property to simplify an expression.

Combine like terms: 2 x2 x22 – 4 x + 10 x – 1 = – 4 x + 10 x – 1 =

2 x2 x22 + (– 4 + 10 ) x – 1 = + (– 4 + 10 ) x – 1 =

2 x2 x22 + 6 x - 1 + 6 x - 1

Evaluating PowersEvaluating Powers 1.21.2

20

Example 1: ( - 3 ) 4 = ( - 3 ) ( - 3 ) ( - 3 ) ( - 3 ) = 81

- 3 4 = - ( 3 3 3 3 ) = - 81

Example 2: Evaluating an algebraic expression

- 3 x 2 – 5 x + 7 when x = - 2

- 3 ( - 2 ) 2 – 5 ( - 2 )x + 7 [ substitute – 2 for x ]

- 3 ( 4 ) – 5 ( - 2 )x + 7 [ evaluate the power, 2 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 2 + n – n 2 = ( 3 n 2 – n 2 ) + n [ group like terms ] = 2 n 2 + 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 ]

Solving Linear EquationsSolving Linear Equations 1.31.3

21

Transformations that produce equivalent equations

Additional property of equality Add same number to both sidesif a = b, then a + c = b + c

Subtraction property of equality Subtract same number to both sidesif 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.

Solve Linear EquationsSolve Linear Equations 1.31.3

22

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 [ subtract 9 from both sides to eliminate the other term ] 3 x = 6 7

7 • 3 x = 7 • 63 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 - 5 n - 5 n [ 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

Solve Linear EquationsSolve Linear Equations 1.31.3

23

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 Add 20 to both sides

x = 1/4 Divide each side by 16

Solve Linear EquationsSolve Linear Equations 1.31.3

24

Equations with fractions

Example 3: 1 x + 1 = x – 1 3 4 6

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:

2 x – 3 y = 62 x – 3 y = 6

– – 3 y = – 2 x + 63 y = – 2 x + 6

y = y = 22 x – 2 x – 2 33

Solve for the formula for the area of a trapezoid for h:

A = A = 11 ( b ( b11 + b + b22) h) h 22

2 A = ( b2 A = ( b11 + b + b22) h) h

2 A2 A = h = h( b( b11 + b + b22))

25

ReWriting Equations and FormulasReWriting Equations and Formulas 1.41.4

ReWriting an Equation with more than 1 variableReWriting an Equation with more than 1 variable 1.41.4

26

Solve : 7 x – 3 y = 8 for the variable y.

7 x – 3 y = 8 - 7 x - 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 ]

x y = 1 – x x x [divide by x to get y by itself ]

y = 1 – x when x = - 1, then y = - 2 and when x = 3, then y = - 2/3 x

Common FormulasCommon Formulas 1.41.4

27

Distance D = r t d = distance, r = rate, t = time

Simple interest I = p r t 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 = ½ b h A = area, b = base, h = height

Area of a Rectangle A = l w A = area, l = length, w = width

Perimeter of Rectangle P = 2 l + 2 w 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 = π r2 A = area, r = radius

Circumference of Circle C = 2 π r C = circumference, r = radius