Ex: Determine if the following are true or false. Ex: Determine whether 4 is a solution of x + 6 = 10

Transcription

1 21 Solving Equations Using Properties of Equality True and False Equations Ex: Determine if the following are true or false 1) True 2) False 3) x Sometimes Sometimes x 4 : True False Ex: Determine whether 4 is a solution of x X T Properties of Equality Let s assume that Thor and Loki are the same height: If they stand together on a box, they are still the same height: So: substitute 4 for Enjoy # A B k (Addition Property of Equality) etnatgus Ff a At C B + c If they stand together in a ditch, they are still the same height: IE 1 T er So: A c B c (Subtraction Property of Equality)

2 5 If you clone Thor and Loki, then they are still the same height: So: k (Multiplication Property of Equality) Hard to draw a picture of: Division Property of Equality Solving Equations: the I # w 0 a ^ Ac Bc 0 00 Be to where 1) Locate the variable(s) and decide which side to keep them on they c are happier 2) Use Addition and Subtraction to move terms so that all the variables are on one side, and all the constants are on the other side 3) Use Multiplication or Division to isolate the variable Ex: Solve L get it all by itself 1) 5t 40 ' 5 t 40 s 4 t 8 2) 80 5w to :# F o ; Este +80 ; +80 W 580 who / 5 8 or W 16

3 i :t 3) s 109 ZS S : 545 : 4) 2a + 4 a + 8 TUYET ; a 4 12 substitute plugin Check : or 2 at 4 at 8 2C 4) +4 E # 5)! # 1 ' # + (" " () () Clear the fractions Denominators : 5, 1, 10,10 LCD 10 The smallest big # they all into go, z t,# F ; Pot +,, a t 8t 3t Check : 5 3 t t T 25 2 to t5d

4 }+ 22 More About Solving Equations Ex: Solve and Check a a 1) 9 (t + 2) 6 (t 3) + 15t ,st 't sii Check pikoytt 's :t3< 9 (3+2) 't 6 (33) IE?sIID 's 1818 True Note: These types of equations are called identities ~~ At ) 5 (x + 3) 3x 2 (x + 8) ZX Solutions : All 9( ot 2) Zx l5#false No Real or Solution No check ( for you ) Numbers R 982 I 6 (03)+156) c Note: These type of equations are called contradictions

5 Clearing Fractions and Decimals Ex: Solve and Check At 1) 0105x + 006(20,000 x) 1740 lsx Check : /20,00012,000) %fs to#nyo& izoo ( 8,000 ) X 12,000 Or Clear decimals 2) " +, + 3, ( ', (( 6,3,9 LCDl8 3nHF nt t + is 5# 3 5h +18 3n 6th ) 241 ) 15h + 54n 69N 6h 6h + 6h + 6h / N 22 * 753,1 n

6 D 1740 w 3 decimal Places 2 decimal Places Worst 3 decimal places Use 10003Zeros Gt ) 10% ( D 1,740, ,200,0001,740, ,200,00060 X 1,740,000 1,200, , *0 X ,200,000

7 ' 55 Check for # 2 It n+3n t # t Et +3t e t a± In LCD9 E *! ( CD Fires I*i ie : Is IF '} _szs ; 2 } F

8 2 23 Applications of Percent Converting from percents to decimals and back again Ex: Convert each from % to decimal, or from decimal to % 1) 31 % 3 I W 2 2) 130% ' O 3 I why 31 % 31 1 Too 30 3) 008 8% 0 08 W 4) % ur Direct Translation Problems Ex: What number is 5% of 96? x t }of Ex: 102 is 213% of what number? at tote int odds OF nae is n Ex: 31 is what percent of 500?, ± Tf! soo 31 50/0 Foo 3 ' det Foo f X not %

9 Applied Percent Problems Ex: (#32) A guest at the San Antonio Hilton Airport Hotel paid $180 for a room plus 9% city room tax, a 1 ¾ % county room tax, and a 6% state room tax Find the total amount of tax that the guest paid on the room Tax Total Tax % rate 180 ( oo a) (00175)+180/006) or 180 ( ) Ex: (#44) A pearl necklace of former First Lady Jacqueline Kennedy Onassis, originally valued at $700, was sold at auction in 1996 for $211,500 Find the percent increase in the value of the necklace Percent Increase / Decrease 180 (01675) $3015 Slept : find the change / difference 211, ,800 step 2 : Calculate the % using original value the Ex: A pair of shoes that usually sells for $110 was marked down to $88 Find the percent decrease in the price of the shoes " gear #, 02 Zfeohease

10 i : Original $700 New : $211,500 Change # ference 210,800 :$ Formal method : The changekiffuena is what % of the original? 210,800 p %5 PEE' 30, % rae#e*toiiiti Round to nearest whole % 30,114% : Round to one decimal place 30,1143% : ' ' ' ' 2 ' ' ' : 30,11429%

11 24 Formulas What are some examples of formulas? Awl Pl+w+l+w Monthan variable one L hftd Formula: An equation that states a mathematical relationship between two or more variables I Prt Interest Principal * interest rate * time P R C Profit Revenue Cost r c + m D RT Retail price cost + markup Distance travelled Rate (speed) * Time C 5/9 (F 32) Temperature in Celcius 5/9 (Temperature in Fahrenheit 32) P 2l + 2w The perimeter of a rectangle two * the length + two * the width A pr 2 The area of a circle (pi) r 2 V pr 2 h The volume of a cylinder pr 2 h Ex: (#14) Find the markup on a dozen roses if a florist buys them wholesale for $1295 and sells them for $4750 : m 's m m$3455@ m

12 Ex: (#20) Three years after opening an account that paid simple interest of 645 % annually, a depositor withdrew the $3,483 in interest earned How much money was left in the account? IPrt t3yeas ) p? r 645% I $ Ex: (#22) Rose Parade floats travel down the 55milelong parade route at a rate of 25 mph How long will it take a float to complete the route if there are no delays? D RT D RT 5m h Dert 5525 hourst' D 55 miles s R ' 2 T? Ex: (#28) Convert 2,212 C, the temperature at which silver boils, to degrees Fahrenheit Round to the nearest degree CIa( F 32 ) hooted 'T I , ( F 32 ) iii i ef E I 19, 9085 F 160 tl60t1#

13 2 2 hours 2 hours ) min 2 hours + 12 minutes 2 5 hours 2 hours 30 minutes 5 of an hour days 1 day, 02 ( 24 ) hours 1 day, 48 hours

14 48 I day, 48 hours i 1 day, 4 hours, 08/60) min I day, 4 hours minutes

15 Solving for a Specified Variable Ex: Solve P R C for R Ex: Solve! for b

16 Ex: Solve for G )" Ex: Solve B 50 + r (x + y) for y

17 25 Problem Solving Ex: (#46) First Aid A sling is in the shape of an isosceles triangle with a perimeter of 144 inches The longest side of the sling is 18 inches longer than either of the other two sides Find the lengths of each side Ex: (#16) A rock group plans to travel for a total of 38 weeks, making three concert stops They will be in Japan for 4 more weeks than they will be in Australia Their stay in Sweden will be 2 weeks shorter than that in Australia How many weeks will they be in each country?

18 Ex: (#40) The three numbers of the combination for a lock are consecutive integers, and their sum is 81 Find the combination

19 26 More Problem Solving Investment Problems Ex: (#22) An investor wants to receive $1,000 annually from two investments He has put $4,500 in a money market account paying 4% annual simple interest How much should he invest in a stock fund that pays 10% annual simple interest to achieve his goal? I Prt Money Market P r t I Stock Fund

20 Uniform Motion Problems D R * T Ex: (#34) How long will it take a mother, running at 4 feet per second, to catch up with her toddler, running down the sidewalk at 2 feet per second, if the child had a 5second head start? Catch up R T D Mother Toddler

21 Mixture Problems Ex: (#40) A photographer wishes to mix 2 liters of a 5% acetic acid solution with a 10% solution to get a 7% solution How many liters of 10% solution must be added? Amount Strength Mixture

22 Ex: (# 48) A store sells regular coffee for $8 per pound and gourmet coffee for $14 per pound To get rid of 40 pounds of the gourmet coffee, a shopkeeper makes a blend to put on sale for $10 per pound How many pounds of regular coffee should he use?

23 NumberValue Problems Ex: (#60) A scuba diver, hired by an amusement park, collected $121 in nickels, dimes and quarters at the bottom of a wishing well There were 500 nickels and 90 more quarters than dimes How many quarters and dimes were thrown into the wishing well?

24 27 Solving Inequalities Ex: For each inequality, express the solution set a) in setbuilder notation, b) as an interval, and c) as a graph 1) x > 5 2) x > 5 3) x < 5 4) x < 5 5) 3 < x < 15

25 Loki and The Hulk Loki is shorter than The Hulk: When they re both standing on a platform, Loki is shorter than The Hulk: When they re both standing in a ditch, Loki Is shorter than The Hulk: When you clone them, Loki is still shorter than The Hulk:

26 However, when you clone them and hang them upside down, what happens? Multiplication and Division Properties of Inequality: 5 < 10 What happens if you multiply both sides by 2? Divide by 5? What happens if you multiply both sides by 2? Divide by 5? What s the rule?

27 Ex: Solve for x: 1) 4 < 4(x 2) 2) 4 < 4 (x 2) < 20 3) '9:() "! + 4