Pre-Algebra Chapter 1 Notes

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

In our number system, there is no such number as 5 ! It is really 5 1 We usually leave out the divisor 1 for convenience.

Common Assumptions with Numbers + 5. 01 1 The sign of a number is positive, + There is decimal point is to the right of the number followed by 0 The power of the number is 1 As a whole number it is over 1 When there is a variable instead of a number, the coefficient is 1 When the variable n has no number in front it is assumed to be 1, thus n is really 1n.

Algebraic Expressions 1.1 Algebraic Expressions 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) Identify the variable in the expression 21 + d. (d) Identify the type of expression for 2 + x and 2 + 3 (2 + x is a variable expression and 2 + 3 is a numerical expression) To Evaluate an Expression replace each variable with a number to find a numerical value. Example 1: Evaluate 5n where n = 6, thus 5 n = 5 (6) = 30 Example: Evaluate the expression where m = 5 and n = 6, thus x + 7 = 11 Example 3: Evaluate the expression where m = 5 and n = 6, thus n – m = n – m = 1

1.2 Powers and Exponents 53 53 = 5 ● 5 ● 5 = 125 Power The result of a repeated multiplication of the same number (factor). Exponent The exponent indicates the number of times the base is used as a factor. exponent base 53 53 = 5 ● 5 ● 5 = 125 Power In words Value 121 12 to the first power 121 = 12 (0.5)2 0.5 to the second power, or 0.5 squared (0.5)(0.5) = 0.25 43 4 to the third power, or 4 cubed 4 ● 4 ● 4 = 64 84 8 to the fourth power 8 ● 8 ● 8 ● 8 = 4096 Writ e the product using an exponent a) 13 ● 13 ● 13 ● 13 = 134 The base 13 is used as a factor 4 times b) (0.2) (0.2)(0.2) = (0.2) 3 The base (0.2) is used as a factor 3 times c) n ● n ●n ●n ●n ●n ● = n6 The base n is used as a factor 6 times

Evaluating Powers with variables 1.2 Evaluating Powers with variables Evaluate the expression x4 when x = 0.5 x4 = (0.5) 4 Substitute 0.5 for x = (0.5) (0.5)(0.5)(0.5) Use 0.5 as a factor 4 times = 0.0625 multiply Evaluate the expression when m = 3 a) m2 = 32 = 3 ● 3 = 9 b) m3 = 32 = 3 ● 3 ● 3 = 27 c) m4 = 32 = 3 ● 3 ● 3 ● 3 = 81

Using Powers in Formulas 1.2 Using Powers in Formulas Use the formula for the volume of a cube. V = s3 Write the formula = (20) 3 Substitute 20 for s = 8000 Evaluate Power 20 in 20 in 20 in Find the area of a square with the given side length 9 meters = 9 x 9 = 81 m2 11 inches = 11 x 11 = 121 in2 1.5 centimeters = 1.5 x 1.5 = 2.25 cm2

1.3 Order of Operations A set of rules to evaluate expressions involving more than one operation Order of Operations Operate within grouping symbols first. Work from the inside to the outside. Simplify powers. Multiply and divide from left to right. Add and subtract from left to right. 23 ● 42 = 2 ● 2 ● 2 + 4 ● 4 8 + 16 24 6 (5 – 3) 2 22 4 42 ● 13 + 8 4 ● 4 ● 1 ● 1 ● 1 + 8 16 ● 1 + 8 = 24 Example Simplify: a. 16 + 8 ● 9 Simplify: b. 18 − 8 ÷ 4 Solutions Multiply first, then add 16 + 8 ● 9 + 72 88 Divide first, then subtract 18 − 8 ÷ 4 18 − 2 16

1.2 Grouping Symbols Grouping Symbols Parentheses ( ) and brackets [ ] and fraction bars 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. x + 4 2 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

Using Grouping Symbols 1.3 Using Grouping Symbols Evaluate Grouping Symbols a) 8 (17 – 2.3) = 8 (14.7) Subtract within parentheses = 117.4 Multiply B) 14 + 6 12 - 7 = (14 + 6 ) ÷ (12 – 7) Rewrite fraction as division = 20 ÷ 5 Evaluate within parentheses = 4 Divide c) 5 ▪ [ 36 – (13 + 9 )] = 5 ▪ [ 36 – 22 ) Add within parentheses = 5 ▪ 14 Subtract within brackets = 70 Evaluate the expression when x = 4 and y = 2 1.2 ( x + 3) = 1.2 x + 3 = 3 x – 2 y = 0.5 [ y – ( x – 2 )] = x2 – y = 2 ( x – y )2 =

Exponents and Grouping Symbols 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 KEEP in MIND 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 = 2 ● 2 ● 2 + 4 ● 4 8 + 16 24 6 (5 – 3) 2 22 4

Using a Problem Solving Plan 1.3 Using a Problem Solving Plan SEWING: You buy a pattern and enough material to make 2 pillows. The pattern costs $5. Each pillow requires $3.95 worth of fabric and a button that costs $.75. Find the total cost. Solution Read and Understand You buy one pattern plus fabric and buttons for 2 pillows. You are asked to find the total cost. Make a plan Write a verbal model Total Cost = Cost of Pattern + Number of pillows ▪ Cost of each pillow Solve the Problem Write and evaluate an expression Total Cost = 5 + 2(3.95 + 0.75) Substitute values into verbal model = 5 + 2(4.70) Add within parentheses = 5 + 9.40 Multiply = 14.40 Add Answer: The total cost is $14.40

Real Numbers and Number Operations 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, 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. Origin  -3 -2 -1 0 1 2 3

Real Numbers and Number Operations 1.1 Real Numbers and Number Operations Graph - 4/3, 2.7, • • • Graph - 2, 3 • • Graph - 1, - 3 • •

Real Numbers and Order of Operation 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 Properties of real numbers include the closure, commutative, associative, identity, inverse and distributive properties. 16

Using Properties of Real Numbers 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 ) ( 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 and 1/a Definition of subtraction: a – b = a + ( - b ) Definition of division: a / b = a 1 / b , b  0

Real Numbers and Number Operations 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 • • Quotient of - 24 and 1/3 ?

Real Numbers and Number Operations 1.1 Real Numbers and Number Operations 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 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

Solve Linear Equations 1.1 Solve Linear Equations Identifying Properties – 8 + 8 = 0 ( 3 • 5 ) • 10 = 3 • ( 5 • 10 ) 7 • 9 = 9 • 7 ( 9 + 2 ) + 4 = 9 + ( 2 + 4 ) 12 (1) = 12 2 ( 5 + 11 ) = 2 • 5 + 2 • 11

1.1 Solve Word Problems Operations 43. What is the sum of 32 and – 7 ? 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 ?

1.1 Solve Unit Measures Unit Analysis 8 1/6 feet + 4 5/6 feet = 27 ½ liters – 18 5/8 liters = 8.75 yards ( $ 70 ) = 1 yard ( 50 feet ) ( 1 mile ) ( 3600 seconds ) = 1 second 5280 feet 1 hour

Algebraic Expressions and Models 1.2 Algebraic Expressions and Models Order of Operations First, do operations that occur within grouping symbols - 4 + 2 ( -2 + 5 ) 2 = - 4 + 2 (3 ) 2 Next, evaluate powers = - 4 + 2 ( 9 ) Do multiplications and divisions from left to right = - 4 + 18 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 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.

Algebraic Expressions and Models 1.2 Algebraic Expressions and Models Example: You can use order of operations to evaluate expressions. Numerical expressions: 8 (3 + 42) – 12  2 = 8 (3 + 16) – 6 = 8 (19) – 6 = 152 – 6 = 146 Algebraic expression: 3 x2 – 1 when x = – 5 3 (– 5 )2 – 1 = 3 (25) – 1 = 74 Sometimes you can use the distributive property to simplify an expression. Combine like terms: 2 x2 – 4 x + 10 x – 1 = 2 x2 + (– 4 + 10 ) x – 1 = 2 x2 + 6 x - 1

1.2 Evaluating Powers 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 7 x + 4 x = ( 7 + 4 ) x [ distributive ] = 11 x [ add coefficients ] 3 n 2 + n – n 2 = ( 3 n 2 – n 2 ) + n [ group like terms ] = 2 n 2 + n [ combine like terms ] 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 Equations 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.

Solve Linear Equations 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 [ subtract 9 from both sides to eliminate the other term ] 3 x = 6 7 • 3 x = 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 - 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 Equations 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 – 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 Equations 1.3 Solve Linear Equations Equations with fractions Example 3: 1 x + 1 = x – 1 3 4 6

ReWriting Equations and Formulas 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: 2 x – 3 y = 6 – 3 y = – 2 x + 6 y = 2 x – 2 3 Solve for the formula for the area of a trapezoid for h: A = 1 ( b1 + b2) h 2 2 A = ( b1 + b2) h 2 A = h ( b1 + b2)

ReWriting an Equation with more than 1 variable 1.4 ReWriting an Equation with more than 1 variable 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

1.4 Common Formulas 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