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9 – 7 - 2011 Do Now 1.) You work 8 hours and earn $60. What is your earning rate? (Important to include units.) $7.50 per hr 2.) You buy 14 gallons of gasoline at $ 3.65 per gallon. What is your total cost? $51.10 3.) A baseball travels at 90 kilometers per hour. What is the speed in meters per second? 25 m / s 1hr 1 min 25m 90 km 1000 m 1km 60 min 60 sec sec 1hr 9 – 8 - 2011 Do Now Use your calculator to evaluate the expression. 1.) 2 59 6 2.) (2) 5 9 6 3.) (144 9) 4.) 75 6 2 3 2 9 – 9 - 2011 Do Simplify the expression. 1.) 4x 3y 2x 5 y 2.) 6( x 4 y ) 2(3x 2 y ) 6 x 24 y 6 x 4 y 6x 2 y 28 y HELPS YOU WITH YOUR ASSIGNMENT! CAN USE THEM ON YOUR TESTS! 1.1 Real Numbers and Number Operations What you should learn: Goal 1 Use a number line to graph and order real numbers. Goal 2 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. positive #’s negative #’s origin -3 -2 -1 0 1 2 3 Graph the numbers on the number line. Ex 1) Ex 2) 0, -3, 1 2, -1, -2 -3 -2 -1 0 1 2 3 Write two inequalities that compare the numbers. Ex 1) Ex 2) 0, -3, 1 2, -1, -2 -3 < 0 <1 -2< -1< 2 Write numbers in increasing order. Ex 1) 0.34, -3.3, 1.12 -3.3 < 0.34 < 1.12 Ex 2) 2.23, 2.2, -2.23 -2.23< 2.2 < 2.23 Properties of Addition and Multiplication 1. a + b = b + a ab=ba Commutative property 2. (a + b) + c = a +(b + c) Associative property (a b) c = a ( b c ) 3. a + (-a ) = 0 1 a 1, a 0 a Inverse property 4. a(b + c) = ab + ac Distributive property 5. a+0=a a 1 a Identity property Using Unit Analysis Perform the given operation. Give the answer with the appropriate unit of measure. example You ride in a train for 175 miles at an average speed of 50 miles per hour. How many hours does the trip take? 175 miles miles 50 hour 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 Do 9 – 12 - 2011 Perform indicated conversion. 1.) 350 feet to yards 116 2 3 yards 2.) 5 hours to minutes 300 minutes 3.) 6800 seconds to hours 1 8 9 hours 4.) 2.2 kilograms to grams 2200 grams 1.2 Evaluate and Simplify Algebraic Expressions What you should learn: Goal 1 Evaluate algebraic expressions. Goal 2 Simplify algebraic expressions by combining like terms. Numerical expression consists of numbers, operations, and grouping symbols. Expressions Containing Exponents. Example: 4 44444 5 The number 4 is the BASE, the number 5 is the EXPONENT, and 4 5 is the POWER. Order of Operations Parentheses Exponents Multiplication and Division Addition and Subtraction 1. First do operations that occur within symbols of grouping. 2. Then evaluate powers 3. Then do multiplications and divisions from left to right 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. Evaluate is to make a substitution, do the work, and determine the value. Value of the expression is the answer after the expression is evaluated. Definitions: Terms: are the number. Coefficient: is the constant in front of the variable. Constant term 4 x y 5xy 25 y 6 y 3 2 Like Terms: Evaluate the power. 4 (2) (2)( 2)( 2)( 2) Ex) Ex) 2 (2)( 2)( 2)( 2) 16 4 16 Evaluate the expression when x = 4 and y = 8. ex) substitute Do the work Get the value 4x 9 y 4(4) 9(8) 16 72 56 Reflection on the Section What does it mean to EVALUATE? assignment 9 – 13 - 2011 Do Now Solve for x. 1.) Perimeter = 45 3x 6 x3 x3 3x 6 x 3 + 3x 6 + 3x 6 + x 3 = 45 1.3 Solving Linear Equations What you should learn: Goal 1 Solve linear equations Goal 2 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 Generalization: If a number has been added to the variable, subtract that number from both sides of the ex) equal sign. x3 6 3 3 If a number has been subtracted from the variable, add that number to both sides of ex) the equal sign. x4 6 4 4 Solving Linear Equations Generalization: If a variable has been multiplied by a nonzero number, divide both sides by that number. example: 4x = - 12 4 4 Solving Linear Equations Generalization: If a variable has been divided by a number, multiply both sides by that number. example: x 6 2 6 6 Hint: you always start looking at the side of the equal sign that has the Variable. Solving Linear Equations 3 2 3 x4 2 3 2 ex) ex) 12 6 x 2 2 5 5 4 x 5 2 2 20 10 x 2 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 example: example: w 10 20 5 10 10 w 5 10 5 5 subtract 10 multiple by 5 4c 2 10 2 2 4c 12 4 4 c3 Solving Multi-Step Equations example: 2x 4 4x 14 6x 4 14 4 4 6 x 18 6 6 x 3 Solving Multi-Step Equations Example) Move the smaller #. 5x 2 3x 8 5x 5x 2 2x 8 8 8 10 2 x 2 2 5 x Solving Linear Equations (Day 1 of 2) Reflection on the Section How does Solving a linear equation differ from Simplifying a linear expression? assignment Page 21 # 3 – 13 odd, 21- 25 odd, 68 Solving Linear Equations example: 2( x 2) 4( x 5) 2x 4 4x 20 4x 4x 6 x 4 20 4 4 6x 24 6 6 x4 Solving Multi-Step Equations 9 – 14 - 2011 Do Perform indicated conversion. 1.) A salesperson at Taylor Ford has a base salary of $20,000 per year and earns a 5% commission on total sales. How much must the salesperson sell to earn $40,000 in one year? $400,000 (Day 2 of 2) example: 3( x 12) 6( x 4) 3x 36 6x 24 6x 6x 9x 36 24 36 36 Book Answer 4 3 Are they the same? 9 x 12 9 9 x 1.33333333 Solving Linear Equations 1 5 example: 40 13 Book Answer 6 2 13 x 13 40 1 8 x2 x2 40 13 x 6.15384615 80 13 Are they the same? Solving Multi-Step Equations Example) 2x 6 2x 8 2x 2x 6 8 No Solution Solving Linear Equations Example) 6x 7 6x 7 6x 6x 77 All Solutions or All Real Numbers work Solving Linear Equations Reflection on the Section What do you do when you have variables on both sides of the equal sign? Page 22 # 33 – 65 odd, 70, 71 Solving Linear Equations 9 – 15 - 2011 Do Now 1.) You can estimate the diameter of a tree without boring through it by measuring its circumference. Solve the formula for d. C = d 1.4 Rewriting Formulas and Equations What you should learn: Goal 1 Rewrite equations with more than one variable Goal 2 Rewrite common formulas. example: Solve for y 2 4 y 10 2 2 4 y 12 4 4 y 3 Rewriting Formulas and Equations example: Solve for y y- = + + Like Terms y= + Rewriting Formulas and Equations example: Solve for y y+ - = y= - + y= - + Rewriting Formulas and Equations example: Solve for y y y y Rewriting Formulas and Equations Solve this equation for y. Ex 1) b ay c b b ay b c a a bc y a (3.7) Formulas 1st Solve this equation for y Ex ) 12 x 3 y 15 12 x 12 x 3 y 12 x 15 -3 -3 -3 y 4x 5 2nd then, find the value of y for the given value of x. (substitute) ; x 2 y 4 ( 2) 5 =3 Formulas Reflection on the Section What does it mean to solve for a variable in an equation? Page 30 # 7 – 14 ALL Page 22 # 56 – 66 EVEN First, solve this equation for y, then substitute. # 7 pg 30 ) 3 x y 26 ;x 7 Formulas Solve this equation for x. Ex 2) 2( x b) c 2x 2b c 2b 2b 2x 2b c 2 2 2b c x 2 (3.7) Formulas Solve this equation for b. Ex 3) 1 A bh 2 h 2 1 h A 1 2 b h 2 1 2A b h (3.7) Formulas Example) Solve the investment-at-simple-interest formula A = P + Prt for t. A = P + Prt -P -P A – P = Prt Pr Pr A-P =t Pr (3.7) Formulas How do you solve for x? 4 x 2x now solve this one for C? s C rC (3.7) Formulas First, substitute the given value for x, then solve this equation for y. Ex ) 12 x 3 y 15; x 2 12(2) 3 y 15 24 24 3 y 9 -3 -3 y 3 Formulas 9 – 16 - 2011 If not NOW, when? You have budgeted $100 to improve your swimming. At your local pool, it costs $50 to join and $5 each visit. Find the number of visits you can have within your budget. Page 1010 # 1 – 27 ALL Page 65 # 1 – 17 ALL No “DO NOW” today Go to STAT 1: Edit *CLEAR the data By highlighting the numbers underneath L1, L2, L3 (not the L1) 1.5 Problem Solving Using Algebraic Models What you should learn: Goal 1 Use general problem solving plan to solve real-life problems Go to STAT 1: Edit *fill in the data Go to STAT again over to CALC 4: LinReg(ax+b) Enter 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. SOLUTION Look at the differences in the heights given in the table. Story Height to top of story (feet) Lobby 1 2 3 4 20 32 32 44 44 56 56 68 After the lobby, the height increases by 12 feet per story. 12 12 12 12 Go to STAT 1: Edit *fill in the data Go to STAT again over to CALC 4: LinReg(ax+b) Enter Look at our data x 0 y 20 1 2 32 44 y = ax+b 3 56 4 68 On your screen a = 12 b = 20 That means your equation is y = 12x + 20 Go to STAT 1: Edit *fill in the data Go to STAT again over to CALC 4: LinReg(ax+b) Enter Page 37 # 11 x 0 y 11 1 2 15 19 y = ax+b 3 23 On your screen a=4 b = 11 That means your equation is y = 4x + 11 Page 37 # 11 – 15, and 22, 23 USING A PROBLEM SOLVING PLAN It is helpful when solving real-life problems to first write an equation in words before you write it in mathematical symbols. This word equation is called a verbal model. The verbal model is then used to write a mathematical statement, which is called an algebraic model. WRITE A ASSIGN WRITE AN VERBAL MODEL. LABELS. ALGEBRAIC MODEL. SOLVE THE ANSWER THE ALGEBRAIC MODEL. 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? Writing and Using a Formula You can use the formula d = rt to write a verbal model. VERBAL MODEL LABELS ALGEBRAIC MODEL Distance = Rate • Time Distance = 550 (kilometers) Rate = r (kilometers per hour) Time = 2.25 (hours) d 550 t = r •(2.25) Write algebraic model. = r Divide each side by 2.25. 244 r Use a calculator. 550 2.25 The Bullet Train’s average speed is about 244 kilometers per hour. Writing and Using a Formula UNIT ANALYSIS You can use unit analysis to check your verbal model. 244 kilometers 550 kilometers hour • 2.25 hours USING OTHER PROBLEM SOLVING STRATEGIES 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 RAILROADS 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 1863. 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? Drawing a Diagram Central Pacific VERBAL MODEL Total miles of track LABELS = Miles per month • Number of months Union Pacific + • Total miles of track = 1590 (miles) Central Pacific rate = 8.75 (miles per month) Central Pacific time = t (months) Union Pacific rate = (miles per month) 20 Union Pacific time = t – 24 ALGEBRAIC MODEL Miles per month 1590 = 8.75 t + 20 (t – 24) Number of months (months) Write algebraic model. Drawing a Diagram ALGEBRAIC MODEL 1590 = 8.75 t + 20 (t – 24) Write algebraic model. 1590 = 8.75 t + 20 t – 480 Distributive property 2070 = 28.75 t 72 = t Simplify. Divide each side by 28.75. The construction took 72 months (6 years) from the time the Central Pacific Company began in 1863. They met in 1869. Drawing a Diagram The construction took 72 months (6 years) from the time The Central Pacific Company began in 1863. The number of miles of track built by each company is as follows: Central Pacific: Union Pacific: 8.75 miles • 72 months = 630 miles month 20 miles month • (72 – 24) months = 960 miles Looking for a Pattern You can use the observed pattern to write a model for the height. VERBAL MODEL LABELS ALGEBRAIC MODEL Height to top of a story = Height of lobby + Height per story • Story number Height to top of a story = h (feet) Height of lobby = 20 (feet) Height per story = 12 (feet per story) Story number = n (stories) h = 20 + 12 n Write algebraic model. = 20 + 12 (15) Substitute 15 for n. = 200 Simplify. The height to the top of the 15th story is 200 feet. 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: Goal 1 Solve simple inequalities Goal 2 Solve compound inequalities. Let’s describe the inequality in different ways. Verbal Phrase All real numbers less than 3 Inequality x<3 Graph -2 -1 0 1 2 3 What about this one… Verbal Phrase All real numbers greater than or equal to 0 Inequality x0 -2 -1 Graph 0 1 2 3 Solving Linear Inequalities You solve these just like you solved other linear equations. Ex 1) x 5 3 5 5 x 2 Subtract 5 Solving Linear Inequalities Ex 2) x 4 7 4 4 x 3 Add 4 Solving 2-Step Linear Inequalities Beware…. 3x 8 2 8 8 3x 6 Watch this… 3 3 Reverse the inequality! Because you divided by a x 2 negative. ex) Solving Linear Inequalities with Variables on both sides 4 x 4 3 5x 5x 5x 9x 4 3 4 4 9x 1 9 9 1 x 9 Solving Linear Inequalities using the Distributive Property 4(2 4 x) 3 8 16x 3 8 8 16 x 5 16 16 5 x 16 Solving Linear Inequalities using Combing Like terms and variables on both sides of the equal sign. 4x 9 4 3 8 3x 4x 5 5 3x 3x 3x x 5 5 5 5 x0 Solving Compound Inequalities Involving “And” 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. 0 x4 -1 0 1 2 3 4 Solve for x. 2 3x 8 10 8 8 8 6 3x 18 3 3 3 2 x6 1 2 3 4 5 6 Solve for x. 2 2 x 1 2 2 2 0 x 3 1 1 1 0 x 3 -3 -2 -1 0 Write an inequality that represents the statement. ex 1) x is less than 6 and greater than 2. 2 x6 ex 2) x is less than or equal to 10 and greater than -3. 3 x 10 x is greater than or equal to 0 and less than or equal to 2. ex 3) 0 x2 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. 85 x 1100 What if... x 3 and x6 What numbers make both statements true? 3 6 What if... x 3 and x6 What numbers make both statements true? No, just the 3 x6 6 What if... x 5 and x 5 Can this happen?? 5 -5 A number can’t be both…. Solving Compound Inequalities Involving “Or” Remember… A Compound Inequality consists of two inequalities connected by the word and or the word or. All real numbers that are Less than -1 or greater than 2. x 1 -2 -1 x2 or 0 1 2 3 Solve for x and graph. 3x 1 4 1 1 3x 3 3 3 x 1 1 or 2x 5 7 5 5 2x 12 2 2 x6 6 Solve for x and graph. x 7 3 7 7 x 10 10 or 2x 24 2 2 x 12 12 Solve for x and graph. x 4 1 4 4 x 5 or 5x 15 5 5 x 3 Is x = -4 a solution? -5 -3 What if... x 3 or x6 What numbers make the statement true? 3 6 What if... x 3 or x6 What numbers make the statement true? All numbers greater than 3 3 6 Reflection on the Section Compare solving linear inequalities with solving linear equalities. assignment 9 –- 2011 Do xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Get Ready! Today We will take the last section of notes when the Quiz C is complete. If you want to retake Quiz B, get one from Mrs. S. I will be checking your Do Nows and Notes (8) Do Nows Notes: 1.1, 1.2, 1.3, 1.4, 1.5, 1.7 1.7 Solving Absolute Value Equations and Inequalities What you should learn: Goal 1 Solve absolute value equations and inequalities An open sentence involving absolute value should be interpreted, solved, and 2 Goal absolute value equations and graphed asUse a compound sentence. inequalities to solve real-life problems. Study the examples:… (8) Do Nows Notes: 1.1, 1.2, 1.3, 1.4, 1.5, 1.7 An open sentence involving absolute value should be interpreted, solved, and graphed as a compound sentence. Study the examples:… ax b c ax b c or ax b c For c 0 , x is a solution of ax b c For c 0 , x has no solution x 2 example 1a) x2 What can x be? or -2 example 1b) x 2 2 x 8 What can x be? Nothing…, no solution x3 4 example 2) x3 4 -3 -3 x 1 -7 or x 3 4 -3 -3 x 7 1 5y 2 7 example 3) 5 y 2 7 or 5y 2 7 +2 +2 +2 +2 5y 9 5 y 5 5 5 5 5 y 1 4 y 1 5 -2 -1 0 1 2 3 example 4) x5 7 2 7 7 1ST get absolute value by itself. x5 9 x5 9 -5 -5 x4 -14 or x 5 9 -5 -5 x 14 4 Reflection on the Section The equation 5x 10 45 has two solutions. Does the equation 5x 10 45 also have two solutions? assignment Page 55 # 9 – 31 odd Page 62 # 17 – 22 Solving Absolute Value Inequalities An absolute-value inequality is an inequality that has one of these forms: ax b c ax b c ax b c ax b c 3x 4 10 example 2) 3x 4 10 -4 and 3x 4 10 -4 -4 3x 14 3x 6 3 3 3 -4 3 2 x 4 3 x2 -5 -4 -3 -2 -1 0 1 2 4x 1 5 example 3) 4x 1 5 -1 4x 1 5 or -1 -1 4x 6 4x 4 4 4 4 4 1 x 1 2 x 1 -3 -1 -2 -1 0 1 2 Solve each open sentence. example 4) 3x 4 7 example 5) 3x 4 7 example 6) 3x 6 0 example 7) 3x 3 0 No solution All numbers work. No solution 1 Graph each on a number line. x 2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x 2 x 2 Let’s do some examples….. 3 2x 5 example 8) 3 2x 5 -3 3 2x 5 and 2x 8 2x 2 -2 -3 -3 -3 -2 -2 x4 x 1 -1 -2 0 1 2 3 4 2x 1 7 example 9) 2 x 1 7 or 2x 1 7 +1 +1 +1 2x 8 2x 6 2 2 2 +1 2 x 3 x4 -3 4 Reflection on the Section How are absolute value inequalities containing a or symbol solved differently from those containing a or assignment symbol?