Chapter 1 An Introduction to Algebra

Transcription

1 Chapter 1 An Introduction to Algebra 1.1 An Introduction to Algebra Symbols Parenthesis/Parentheses Bracket/Brackets Brace/Braces Algebraic Expressions vs. Algebraic Equations Operation Variable Constant What operations does the equation 4y = 5/6 contain? What is the variable? What are the constants?

2 1.2 Fractions Prime number: Composite number: Prime numbers between 1 and 100: Express 1240 as a product of primes

3 Division Tricks: Divisible by 2: Divisible by 3: Divisible by 5: Divisible by 6: Divisible by 9: Divisible by 10: Pizzas and Squirrels

4 Simplify!"# $%& Perform the operations and simplify, if possible 1) % ' & ' 2) % ' & ' 3) % ' + & ' 4) % ' & ' 5) 2 $ ' 3

5 1.3 The Real Numbers Natural Numbers Whole Numbers Integers Rational Numbers Terminating Decimal or Repeating Decimal Irrational Numbers Non-terminating, non-repeating decimal Real Numbers

6 Venn Diagram: Classify the following numbers: {0.1, 45, -2, -13/4, 12/4, -6 7/8, pi, 2, 9, 0 } Graphing, Order and the Real Number Line The Real Number Line Graph these on a number line: {0, 5, -6, 0.3, -1/2, -5 ½, }

7 What is an opposite? Give examples. Inequalities < Less Than > Greater Than Use < or > to make each statement true: 1) 4 5 2) ) ) 4-5 5) ) 0-2 Absolute Value Definition: The absolute value of a number is its difference from zero. 3 = -3 = 0 = - -3 =

8 1.4 Adding Real Numbers, Properties of Addition Modeling Addition on the Number Line Step 1: Start at 0 Step 2: Move 4 units to the right Step 3: Move 5 more units to the right Add Use a graph if you re unsure 1) ) (-2) + 3 3) 2 + (-3) 4) (-2) + (-3) What are the rules of addition? Like Sign Unlike Sign -

9 Properties of Addition The Commutative Property of Addition: a + b = b + a The Associative Property of Addition: (a + b) + c = a + (b + c) Addition Property of 0 (Identity Property): a + 0 = a Addition Property of Opposites (Inverse Property): a + (-a) = 0

10 1.5 Subtracting Real Numbers The Minus Symbol 5 18 is read as Five minus eighteen - 5 is read as Negative 5 -(-5) is read as The opposite of negative 5 Simplify 1) -(-3) 2) -(-(-3)) 3) -(0) 4) ) -(-3)

11 Modeling Subtraction on the Number Line 5 4 Step 1: Start at 0 Step 2: Move 5 units to the right Step 3: Move 4 units to the left (Subtraction tells us to change the direction.) But wait then what s 5 + (-4) =? Fact: Subtraction is adding the opposite. Perform the operations. 1) 5 (-4) 2) -5 (-4) 3) ) -24 (-28)

12 1.6 Multiplying and Dividing Real Numbers Negative numbers and goth kids (baby bats) Multiply 1) (-9) (-3) 2) (-1) (-2)(-3)(-4)(-5) Fact: 1 times any real number is: Fact: 0 times any real number is: Fact: -1 times any real number is: Fact: The product of a nonzero real number and its reciprocal is: Find the reciprocal of each number and then multiply 1) 2 2) -2 3) 2/3 4) -1/5 5) 0

13 Fact: Division comes from multiplication What is 6 divided by 2? Why? Divide and check with multiplication 1) 0 1 2) (-27) 9 Over an 8-year period, the value of a $150,000 house fell at a uniform rate to $110,000. Find the amount of depreciation per year.

14 1.7 Exponents and Order of Operations An exponent is used to indicate repeated multiplication. It tells how many times the base is used as a factor. 2 3 : Two cubed or two to the third power = 3 2 = Three squared or three to the second power = 4 5 = Four to the fifth power ' 1 1 = On a calculator: Negative bases (-2) 2 = (-1) 6 = (-2) 3 = -1 2 = (-2) 4 = -3 3 = (-1) 5 = -2 4 =

15 Order of Operations: 1) Grouping Symbols 2) Exponents 3) Division/Multiplication, Left to Right 4) Subtraction/Addition, Left to Right Evaluate 1) 3 * 4-2 2) ) ) 3 * 2 3 5) -4[2 + 3(8 4 2 ) 2 ] 2

16 6) )! "3% 4 3$ ' 5 3(3$)

17 1.8 Algebraic Expressions Identify the terms and the coefficients of each term in the following expression: 7x 2 x + 6 Declaring Variables Write an expression that represents the area of a square (Really good chart on page 70) Write each phrase as an algebraic expression 1) 13 more than x 2) 13 less than x 3) x less than 13 4) 13 times x 5) The ratio of 13 to x 6) Double x 7) Triple x 8) 8 greater than twice x

18 9) 8 less than twice x Write an expression that represents each situation 1) p pounds of Peanuts were mixed with c pounds of cashews to make 100 pounds of a mixture. 2) How many feet are there in y yards? 3) If one egg is worth g cents, find the value (in cents) of one dozen eggs. 4) The expression 20,000 3s gives the number of square feet of sod that are left in a field after s strips have been removed. Suppose a city orders 7,000 strips of sod. Evaluate the expression and explain the result.

19 1.9 Simplifying Algebraic Expressions Using Properties of Real Numbers Commutative Property of Addition: a + b = b + a Commutative Property of Multiplication: ab = ba Associative Property of Addition: (a + b) + c = a + (b + c) Associative Property of Multiplication: (ab)c = a(bc) New The Distributive Property aka Visiting the Family 4(5 + 3) =? Order of Operations: Breaking up the multiplication into two pieces: Did we get the same answer each time? The Distributive Property: a (b+c) = ab + ac a (b c) = ab - ac Multiply a) 3(4x + 5) b) 3(4x - 5) c) -3(4x - 5)

20 d) 10 : ' + 1 & e) -0.5 (2t w) Combining Like Terms Like terms: Unlike terms: Add 3x + 4x using the Distributive Property Simplify by combining like terms 1) 9z 7 2) 9z 7 z 3) 9z 7 z 19z 4) 43 s 3 44 s 3

21 5) 43 s 4 44 s 3 Simplify 1) x + x 2) x * x 3) x + x + x 4) 2x + x + 5 5) 6x y + 2y 3x 12 6) 3z y 2 + 2y 10z 4y + 3