CAN USE THEM ON YOUR TESTS! HELPS YOU WITH YOUR ASSIGNMENT!

1.1 Real Numbers and Number Operations What you should learn: Goal1 Goal2 Use a number line to graph and order real numbers. Identify properties of and use operations with real numbers.

Whole numbers: 0, 1, 2, 3, … Integers: …-3, -2, -1, 0, 1, 2, 3, … Real numbers: include fractions, decimals, whole numbers, and Integers origin positive #’s negative #’s

Graph the numbers on the number line. Ex 1) , -3, 1 Ex 2) 2, -1, -2

Write two inequalities that compare the numbers. Ex 1) 0, -3, 1 Ex 2) 2, -1, < 0 <1 -2< -1< 2

Write numbers in increasing order. Ex 1) 0.34, -3.3, 1.12 Ex 2) 2.23, 2.2, < 2.2 < < 0.34 < 1.12

Properties of Addition and Multiplication 1. a + b = b + aCommutative property 2. (a + b) + c = a +(b + c)Associative property 3. a + (-a ) = 0 Inverse property a b = b a (a b) c = a ( b c )

4. a(b + c) = ab + ac Distributive property Identity property 5. a + 0 = a

Using Unit Analysis Perform the given operation. Give the answer with the appropriate unit of measure.

Reflection on the Section When converting units with unit analysis, how do you choose whether to use a particular conversion factor or its reciprocal? assignment

1.2 Algebraic Expressions and Models What you should learn: Goal1 Goal2 Evaluate algebraic expressions. Simplify algebraic expressions by combining like terms.

Numerical expression Numerical expression consists of numbers, operations, and grouping symbols. Expressions Containing Exponents. Example: The number 4 is the BASE, the number 5 is the EXPONENT, and is the POWER.

Order of Operations Parentheses Exponents Multiplication and Division Addition and Subtraction 3. Then do multiplications and divisions from left to right 1. First do operations that occur within symbols of grouping. 2. Then evaluate powers 4. Finally do additions and subtractions from left to right.

Variable is a letter that represents a number. Values of the variable are the numbers. Algebraic expression is a collection of numbers, variables, operations, and grouping symbols. Value of the expression is the answer after the expression is evaluated. Evaluate is to make a substitution, do the work, and determine the value.

Definitions: Terms: are the number. Coefficient: is the constant in front of the variable. Like Terms: Constant term

Ex) Evaluate the power.

Evaluate Evaluate the expression when x = 4 and y = 8. ex) substitute Do the work Get the value

Reflection on the Section State the order of operations. assignment

1.3 Solving Linear Equations What you should learn: Goal1 Goal2 Solve linear equations Use linear equations to solve real-life problems.

Using Addition or Subtraction The key to success: Whatever operation is done on one side of the equal sign, the same operation must be done on the other side. Inverse operations undo each other. Examples are addition and subtraction. Solving Linear Equations Solving Linear Equations

Generalization: If a number has been added to the variable, subtract that number from both sides of the equal sign. If a number has been subtracted from the variable, add that number to both sides of the equal sign. ex) Solving Linear Equations Solving Linear Equations

Generalization: If a variable has been multiplied by a nonzero number, divide both sides by that number. example: 4x = Solving Linear Equations Solving Linear Equations

Generalization: If a variable has been divided by a number, multiply both sides by that number. example: Hint: you always start looking at the side of the equal sign that has the Variable. Solving Linear Equations Solving Linear Equations

ex) Solving Linear Equations Solving Linear Equations

Generalization: First undo the addition or subtraction, using the inverse operation. Second undo the multiplication or division, using the inverse operation. Solving Linear Equations Solving Linear Equations

example: subtract 10 multiple by 5 Solving Multi-Step Equations

example: Solving Multi-Step Equations

example: Solving Linear Equations Solving Linear Equations

example: Solving Multi-Step Equations

Example) Move the smaller #. Solving Linear Equations Solving Linear Equations

Example) No Solution Solving Linear Equations Solving Linear Equations

Example) All Solutions or All Real Numbers work Solving Linear Equations Solving Linear Equations

Reflection on the Section How does solving a linear equation differ from simplifying a linear expression? assignment Solving Linear Equations Solving Linear Equations

1.4 Rewriting Equations and Formulas What you should learn: Goal1 Goal2 Rewrite equations with more than one variable Rewrite common formulas.

Solve this equation for x. Ex 2) (3.7) Formulas

Solve this equation for x. Ex 3) 22 (3.7) Formulas

Solve this equation for b. Ex 5) hh (3.7) Formulas

Example) Solve the investment-at-simple-interest formula A = P + Prt for t. A = P + Prt -P A – P = Prt Pr = t A - P Pr (3.7) Formulas

How do you solve for x? now solve this one for C? (3.7) Formulas

First, substitute the given value for x, then solve this equation for y. Ex ) -3 Formulas

First, solve this equation for y, then substitute. Ex ) -3 Formulas = 3

Reflection on the Section How can rewriting formulas help you solve them? assignment

1.5 Problem Solving Using Algebraic Models What you should learn: Goal1 Goal2 Use general problem solving plan to solve real-life problems Use other problem solving strategies to help solve real-life problems.

This word equation is called a verbal model. U SING A P ROBLEM S OLVING P LAN The verbal model is then used to write a mathematical statement, which is called an algebraic model. W RITE A VERBAL MODEL. A SSIGN LABELS. W RITE AN ALGEBRAIC MODEL. It is helpful when solving real-life problems to first write an equation in words before you write it in mathematical symbols. S OLVE THE ALGEBRAIC MODEL. A NSWER THE QUESTION.

Writing and Using a Formula The Bullet Train runs between the Japanese cities of Osaka and Fukuoka, a distance of 550 kilometers. When it makes no stops, it takes 2 hours and 15 minutes to make the trip. What is the average speed of the Bullet Train?

r = Write algebraic model. Divide each side by Use a calculator. r 244  Writing and Using a Formula L ABELS V ERBAL M ODEL Distance = Rate Time 550Distance = (kilometers) 2.25Time = (hours) rRate = (kilometers per hour) A LGEBRAIC M ODEL You can use the formula d = r t to write a verbal model. The Bullet Train’s average speed is about 244 kilometers per hour. d = rt r550(2.25) =

Writing and Using a Formula You can use unit analysis to check your verbal model. 550 kilometers  244 kilometers hour 2.25 hours UNIT ANALYSIS

U SING O THER P ROBLEM S OLVING S TRATEGIES When you are writing a verbal model to represent a real-life problem, remember that you can use other problem solving strategies, such as draw a diagram, look for a pattern, or guess, check and revise, to help create a verbal model.

Drawing a Diagram R AILROADS In 1862, two companies were given the rights to build a railroad from Omaha, Nebraska to Sacramento, California. The Central Pacific Company began from Sacramento in Twenty-four months later, the Union Pacific company began from Omaha. The Central Pacific Company averaged 8.75 miles of track per month. The Union Pacific Company averaged 20 miles of track per month. The companies met in Promontory, Utah, as the 1590 miles of track were completed. In what year did they meet? How many miles of track did each company build?

Write algebraic model. 1590=8.75+(t – 24)20t Union Pacific time = (months) t – 24 Union Pacific rate =20 (miles per month) Central Pacific time = (months) t Central Pacific rate =8.75 (miles per month) Total miles of track =1590 (miles) Drawing a Diagram A LGEBRAIC M ODEL L ABELS V ERBAL M ODEL Total miles of track = + Number of months Miles per month Central Pacific Number of months Miles per month Union Pacific

Divide each side by = t The construction took 72 months (6 years) from the time the Central Pacific Company began in They met in =8.75t+20(t – 24) A LGEBRAIC M ODEL Write algebraic model. Drawing a Diagram 1590 = 8.75 t + 20 t – = t Distributive property Simplify.

Drawing a Diagram The number of miles of track built by each company is as follows: Central Pacific: Union Pacific: 72 months (72 – 24) months 8.75 miles 20 miles month = 630 miles = 960 miles The construction took 72 months (6 years) from the time The Central Pacific Company began in 1863.

12 Looking for a Pattern The table gives the heights to the top of the first few stories of a tall building. Determine the height to the top of the 15th story. After the lobby, the height increases by 12 feet per story. S OLUTION Look at the differences in the heights given in the table. Story Height to top of story (feet) Lobby

You can use the observed pattern to write a model for the height. Substitute 15 for n. Write algebraic model. Simplify. = + h2012n Height to top of a story =h (feet) Height per story =12 (feet per story) Height of lobby =20 (feet) A LGEBRAIC M ODEL = 200 Height to top of a story = Height per story Story number Height of lobby + = (15) Story number =n (stories) L ABELS V ERBAL M ODEL The height to the top of the 15th story is 200 feet. Looking for a Pattern

Reflection on the Section After you have set up and solved an algebraic model for problem description, what remains to be done? assignment

1.6 Solving Linear Inequalities What you should learn: Goal1 Goal2 Solve simple inequalities Solve compound inequalities.

Verbal Phrase All real numbers less than 3 Inequality x < 3 Graph Let’s describe the inequality in different ways.

Verbal Phrase All real numbers greater than or equal to 0 Inequality Graph What about this one…

Solving Linear Inequalities Ex 1) You solve these just like you solved other linear equations. Subtract 5

Solving Linear Inequalities Ex 2) Add 4

Solving 2-Step Linear Inequalities ex) Beware…. Watch this… Reverse the inequality! Because you divided by a negative.

Solving Linear Inequalities with Variables on both sides

Solving Linear Inequalities using the Distributive Property

Solving Linear Inequalities using Combing Like terms and variables on both sides of the equal sign.

Solving Compound Inequalities Involving “And” and or A Compound Inequality consists of two inequalities connected by the word and or the word or.

All real numbers that are greater than or equal to zero and less than 4. All real numbers that are greater than or equal to zero and less than

Solve for x

0 -3-2

Write an inequality that represents the statement. ex 1) x is less than 6 and greater than 2. ex 2) x is less than or equal to 10 and greater than -3. ex 3) x is greater than or equal to 0 and less than or equal to 2.

Write an inequality that represents the statement. ex 4) The frequency of a human voice is measured in hertz and has a range of 85 hertz to 1100 hertz.

What if... and 3 6 both What numbers make both statements true?

What if... and 3 6 both What numbers make both statements true? No, just the

What if... and -5 5 Can this happen?? A number can’t be both….

Solving Compound Inequalities Involving “Or” Remember… andor A Compound Inequality consists of two inequalities connected by the word and or the word or. Remember… andor A Compound Inequality consists of two inequalities connected by the word and or the word or.

All real numbers that are or Less than -1 or greater than 2. or

Solve for x and graph. or 61

or 1210

Solve for x and graph. or -3-5 Is x = -4 a solution?

What if... or 3 6 What numbers make the statement true?

What if... or 3 6 What numbers make the statement true? All numbers greater than 3

Reflection on the Section Compare solving linear inequalities with solving linear equalities. assignment

1.7 Solving Absolute Value Equations and Inequalities What you should learn: Goal1 Goal2 Solve absolute value equations and inequalities Use absolute value equations and inequalities to solve real-life problems.

absolute value compound An open sentence involving absolute value should be interpreted, solved, and graphed as a compound sentence. Study the examples:…

For, x is a solution of if x is a solution of: or For, x has no solution

example 1a) or 2 -2 example 1b) What can x be? Nothing…, no solution

example 2) or

example 3) or

example 4) or ST absolute value get absolute value by itself.

Solving Absolute Value Inequalities An absolute-value inequality is an inequality that has one of these forms:

example 2) and

example 3) or

example 4) Solve each open sentence. example 5) No solution All numbers work. example 6) No solution example 7) 1

Graph each on a number line Let’s do some examples…..

example 8) and

example 9) or

Reflection on the Section How are absolute value inequalities containing a assignment symbol solved differently from those containing a or symbol?