No More Double Signs Page 1
The Bracket Method: there is a simple way to learn algebra 2009-2010
No More Double Signs Page 2
The Bracket Method: there is a simple way to learn algebra 2009-2010
No more double signs
A โ+โ, or โ-โ sign can mean many things; create a list of words for both signs:
โ+โ: positive, plus, add, _________________________________________________________________
_____________________________________________________________________________________
โ-โ: negative, subtract, minus, ____________________________________________________________
_____________________________________________________________________________________
Circle all double signs and replace with one sign, then punch the problem in your calculator exactly as you see it!
1: 51 + -36 โ 46 + 71 + -26 2: 15 + 62 - +18 - -12 - 6
What happens when the first number is โ-โ? You must tell the calculator the starting point is negative! The only
time you use the โnegative signโ when adding or subtracting is to tell the calculator the first number is negative.
3: -32 - +65 โ 90 - -61 + 17 4: -28 โ 18 + 92 + -28 + -65
5: -120 โ 85 + -62 - +24 - 82 6: 18 โ 9 + -68 + 25 โ 98 - -8 - +89
7: 87 - -68 + 54 + -8 - -12 - +37 8: -61 + -72 - 6 - -35 - +74
The Big Ugly (TBU): -125.89 + -95.32 + 58.6 - -71.6 + 36.9 - 17.68 - +85.13 + 98.2 - -84.2
TI-83
The colored buttons on your calculator are math operations (add and subtract). Use these
buttons only! Do not touch the negative sign if you want to add or subtract!
TI-83
The negative (-) button is only used to tell the calculator the starting point of an addition or
subtraction problem is negative. If the first number is negative, use this button.
Add
Subtract
Negative
No More Double Signs Page 3
The Bracket Method: there is a simple way to learn algebra 2009-2010
No More Double Signs
Everything in math is either positive or negative; the sign of the number tells you both: (1) if it is positive or
negative; and (2) to add or subtract when combining. No more โ5 + -10โ or โ6 - +7โ. A number is either + or โ!
Write the following real life scenarios as a mathematical expression with one sign in front of each number
1: balance of your checking account is $134 2: write a check for $103 3: the temperature drops by 16 degrees, then falls another 6 degrees. 4: you sell $350 at a garage sale 5: you lose 5 points on an exam for not writing your name, but you got the extra credit right for 2 extra points 6: 60 students left the concert early. 23 students came in late. 7: the fence has to be 7 feet longer A little tougherโฆremember one sign for each number! 9: Starting temperature is 67 degrees; the temperature rises 15 degrees at noon, then goes up 8 more degrees by dinner time; the temperature then drops 19 degrees at sundown, and decreases another 5 during the night. b: What is the final temperature? 10: The business account had a starting balance of $1340. You write a check for $245; then you take $160 out at an ATM; then you make a deposit for $378; finally, you write another check for $29.
b) What is the final balance?
Speak Geek
Expression: numbers and/or variables put together in a mathematical sentence with + or โ signs (no equal sign).
No More Double Signs Page 4
The Bracket Method: there is a simple way to learn algebra 2009-2010
No more double signs
Find the Mistakes:
1: 51 + -36 โ 46 + 71 + -26 2: 15 + 62 - +18 - -12 โ 6 3: 18 - 52+ -53 - -12 โ 97 + -31
51 - 36 - 46 โ 71 โ 26 15 + 62 โ 18 โ 12 โ 6 18 โ 52 + 53 + 12 โ 97 + 31
Prove the Commutative property by going through the following examples. a & b are the same problem, just a different order. Put both in the calculator; see if you get the same answer.
4a: -32 - 65 โ 90 + 61 + 17 5a: -120 + 85 - 62 + 24 โ 82 6a: -14 + 75 - 12 - 34 + 45
4b: 61 โ 32 + 17 โ 65 โ 90 5b: 85 โ 62 - 82 + 24 โ 120 6b: 45 โ 12 - 34 + 75 - 14
Now that you know the Commutative Property works: find the mistakes; why donโt these problems work?
7a: -120 โ 85 + 62 - 24 + 82 8a: -14 + 75 - 12 - 34 + 45 9a: -15 โ 5 + 82 - 24 + 59
7b: -85 + 62 + 82 + 24 -120 8b: 45 โ 12 + 34 โ 75 + 14 9b: -24 - 15 + 59 + 5 + 82
Fraction Time: Get rid of double signs; Use your calculator to find the answer; put the answer as a fraction.
10:
3
5+
7
9โ
โ1
4 +
4
7+
โ6
7
11: โ3
5 โ
+1
8โ
โ3
4 +
โ5
6+
3
7
TBU: 3
5+
โ7
9โ
+1
4 +
4
7โ
โ6
7
TI-83 A fraction is a division problem in disguise. To enter a fraction, simply type the top number, โรทโ, and the bottom number. To be safe, put all fractions inside (); do not put the operation sign inside the ()
example: (3รท5) โ (5รท7)
Donโt worry about simplifying or reducing your answers, the calculator can do that: Press [Math], [1], [Enter]. You should be able to change any answer from decimal to fraction with the press of 3 buttons: โMath, one, enterโ.
Speak Geek
Commutative Property: Do you see the โCoโ in the word Commutative? Remember Co -โchange orderโ. It says
you can change the order of an addition (or subtraction) problem, without changing the order.
1: [Math]
2: [1]
3: [Enter]
How to Bracket Page 5
The Bracket Method: there is a simple way to learn algebra 2009-2010
How to Bracket Page 6
The Bracket Method: there is a simple way to learn algebra 2009-2010
How to Bracket
Bracketing Terms: the first step to the bracket method is to be able to โbracket the termsโ in an expression
What is the rule for bracketing:_____________________________________________________________
______________________________________________________________________________________
Bracket the following expressions
1: 5x2 โ 6(3x) + 5y(8x) - 17 2: -6x(4y) โ 7x(2y-7t) + 6z โ 8(5) + 19(-t)
3: -2(5x) + 6(2x-7+8y) + 9(3)(-7) 4: 6x2(4x) โ 2(5y)(-6)(-9y) โ 6t โ 7t3 + 2
5: -5 + 6y โ (5)(-8x)(6y) - 8t(-6x)(3y4) โ 4(-7x) โ (-x)
Donโt panic, stick with your rules!
6: 5 6 + 7 โ 3๐ฅ โ 8 7๐ฅ โ (2!)(6๐ฅ โ 5๐ฆ) 7: 5๐ฅ 7๐ฆ โ 6๐ฅ2 + sin 6๐ฅ โ cos(3๐ฅ โ 9)
TBU: -6(5x) + 6x(6y)(-6) โ (4x +5-7y) โ 8(4)(-6) + 5x โ 6(-x)(-y) + 4 โ 6(-8) โ 2 + 9x(-3x)(-6x)
How well have you trained your eyes? Without bracketing, write the number terms in each expression?
8: 5x(-40) โ 7(-3) โ (3x-4) โ (3-6y) 9: 4(3x)(4y)(-5z) โ 2(3) โ (53y) โ 2(-5)
10: -6(3x)(-6y) โ 6x โ 6(4x-7) - 6(7) 11: -5x(7y)(-6) โ 2x2 + 5(-3x-2y2) โ x(5y) + (-t)
Speak Geek
Term: a part of a mathematical expression or equation separated by + or โ signs.
How to Mash Page 7
The Bracket Method: there is a simple way to learn algebra 2009-2010
How to Mash Page 8
The Bracket Method: there is a simple way to learn algebra 2009-2010
Multiplication = Mash potatoes
In algebra, the easiest operation is multiplication. Just put it all the signs, numbers, letters in one pile (like your
uncleโs mash potato plate at thanksgiving: potatoโs, butter, peas, carrots, etc.),
Multiply each term
1: 4(3x)(6y) 3: 7x(7y)(5t)(2) 4: 6(2x)(5)(2t)(10) 5: x(y)(t)(z) 6: 2(x)(5)(2y)
Donโt forget about signs? (In multiplication and division, every 2 negative signs is equal to a positive!)
Even # of โ signs = _____ Odd # of โ signs = _____ Multiply each term: (1) signs; (2) numbers; (3) letters
7: -3(4x)(7y)(-5t) 8: 4t(6y)(-5)(-8z) 9: -6(-4x)(-7y)(5t)(10z)
10: 4(-3x)(12y)(-4)(-9z) 11: -(2)(3x)(-7y)(-10)(-8t) 12: -x(y)(-z)(t)
13): -6.8x(2.3y)(-7.1z) 14: 2.5(-6.8x)(6.25y)(-6.5t) 15: -(3.4x)(8.4y)(-9.2t)(-4)
TBU: 3
5 7๐ฅ โ
3
4
2
8๐ฆ โ5๐ก โ๐ก (โ8๐ง)
Math Geeks Only:
16: Is 5(4)(-6)(-10) the same as -6(5)(-10)(4)?
17: What property allows you to change the order and keep the same answer? (ahemโฆchange order)
How to Mash Page 9
The Bracket Method: there is a simple way to learn algebra 2009-2010
Bracket and Mash
Each term (bracket) is a potential multiplication (mash) group. After you bracket, mash each term.
Bracket and then mash each term separately (signs, numbers, letters)
1: 4(3x) - 4(-5x)(2y) - 3(4y) 2: -5x(4y)(2) โ 5(-2x) + 7(3t)(-5x)
3: 3(-5) + 3x(-7y) - 3(-x)(y)(-t) 4: -(4)(-5) โ 5(-2x)(-7) + 4(-6x)(y)
5: -(-4x) + (-6)(-5t)(7z) โ x(-y)(-z) 6: 7(3) โ 4(-8x) + 4x(2y) โ 6(-8t)
7: x โ 4t(6)(3) โ 6 + 2x(-6) 8: -6(-4)(-5)(7) โ (2x)(3y)(-6y)3
9: -(-1)(-1)(-5) + 3x(-3z) + 5(-y) 10: 3.2(-6.5) + 4x(-5.7)(-2.1) โ 3.6z(-4.5t)
11: -12(7x)(-15y) โ 5(-19)(6t) โ (-12x)(-25) 12: -t โ (-x)(-t)(-y) โ (v)(-z)(-t) โ x(z)
TBU: 3
5๐ฅ 7๐ฆ โ 5
2
3๐ฅ โ
4
7๐ก + 5.6๐ก โ5๐ฅ
2
5๐ฆ โ โ7.2๐ก + 5.8
7
9๐ฆ
6
5๐ก โ ๐ฅ โ๐ฆ โ8 +
9
3
How to Mash Page 10
The Bracket Method: there is a simple way to learn algebra 2009-2010
Mixed Review
Part I: No more double signs โ you know what to do!
1: -32 - +65 โ 90 - -61 + 17 2: -28 โ 18 + 92 + -28 + -65
3: -120 โ 85 + -62 - +24 - 82 4: 18 โ 9 + -68 + 25 โ 98 - -8 - +89
Part II: Bracket the following terms โ donโt do the math, just see how many terms there are!
5: 5(2x) โ 7(-3) + x(y)(-z) 6: sin32 โ 14(x) + 4cos(3x) โ 2(3)(4)
7: 3
4 4x-7 + 3 49-5x +
2
5 x (y) 8: 4 โ (7 -2x) โ (x)(2y) + 17(4) โ 2x3 + 16(-x2)(x)3
Part III: Mash the following terms
9: 2(-3x)(-6y) 10: -(-3x)(4y)(-7z) 11: -2(-x)(-7)(-y)
12: 7(-6y)(5x)(-t) 13: 3
5 5๐ฆ
2
3๐ก (โ
3
8๐ฅ) 14: -(-x)(-y)(-t)(-z)
Part IV: Bracket terms; mash each term.
15: 2(-4) + 4x(-6y) - 3(-x)(5)(-t) 16: -(3)(-5) โ 2(-3x)(-5) + 3(-6x)(y)
17: -(-4x) + (-3)(-2t)(7z) โ x(-2)(-z) 18: 5(4) โ (-8z) + 4x(2y) โ 3(-7t)
How to Distribute (Pizza Delivery) Page 11
The Bracket Method: there is a simple way to learn algebra 2009-2010
How to Distribute (Pizza Delivery) Page 12
The Bracket Method: there is a simple way to learn algebra 2009-2010
How to Distribute (Pizza Delivery) Page 13
The Bracket Method: there is a simple way to learn algebra 2009-2010
Distribute: the second type of multiplication
Write M (mash) or D (distribute) above each termโฆthen do it!
1: 6(3x)(-5) 2: 7(5x + 7)
3: 7(x + 9) 4: -5(6 โ x + 7y)
5: -(6x)(+7) 6: -(7 โ 5x)
7: 5(4x)(7y)(6z) 8: 3(5x-8y+9)
Distribute each problem (draw those arrows if you are not sure!)
9: 6( 4x + 8) 10: -5(7 + 3t)
11: -7(6 โ 8x) 12: 2y( -7 โ 4x)
13: - ( 5x โ 8) 14: 5(t โ x)
15: -5(2x โ y + 5) 16: (6 โ 5x + 7y)
TBU1: -2xy(6t โ 9 + 4z โ u) TBU2: 3
4(5๐ฅ โ
1
3๐ฆ +
5
9๐ง)
How to Distribute (Pizza Delivery) Page 14
The Bracket Method: there is a simple way to learn algebra 2009-2010
Mixed Review
Part I: No more double signs โ you know what to do!
1: -22 - +61 โ 23 - -16 + 37 2: -18 โ 38 + 52 + -8 + -25
3: -12 โ 65 + -75 - +44 - 62 4: 8 โ 19 + -38 + 27 โ 81 - -15 - +64
Part II: Bracket the following terms โ donโt do the math, just see how many terms there are!
5: 3(-2x) โ 5(-2) + x(3y)(-4) 6: 32(7!) โ (2x) + tan(3x) โ 2!(3)(4)
7: -3 49 +3
4 4x + ๐ฅ2 4(๐ฆ)3 8: โ (y -2x) โ (-x2)(-18x)3 - (5x)(3y) + 17(4) โ 2x3(5x)
Part III: Bracket terms; mash each term
9: 2(5x) - 4(-x)(2y) - 8(5y) 10: โ 6(-2x) - 5x(4y)(2) + (4t)(-7x)
11: -(-7) - 2(-x)(y)(-7) + 3x(-7y) 12: -(3)(-2)(7)(v) โ 5(-2x)(-7t)(-w) + 4(-6x)(y)+(2)
Part IV:. Mash or Distribute the following terms
13: 6(3x)(-5) 14: 7(5x + 7) 15: -(4x2 โ 5x โ 8)
16: 2
3(
1
2๐ฅ โ
1
3) 17: -(-4x)(-7)(-6y) 18: -5(6 โ x + 7y)
PEMD: The Bracket Way Page 15
The Bracket Method: there is a simple way to learn algebra 2009-2010
PEMD: The Bracket Way Page 16
The Bracket Method: there is a simple way to learn algebra 2009-2010
Bracket and Identify
Find Terms (Bracket), then write M (mash) or D (distribute) or S (solo) above the bracket; do not do the math!
1: 4(2x) โ 3(2 +6y) 2: -2(5y) + 4(2)(5) โ 3(2x)
3: -3(4 -5x) โ 2(4y) + 2(3t)(-z) 4: 5(3x-6y) โ 4(-5)(-3) โ 4(t)
5: -(4x)(-5) โ (5y โ 7) 6: -3(2x) โ 4(5 โ 3y) โ 4w + 2(3xy โ 5t) Put it all together now: bracket, decide and label each term (M, D or S); do the math one term at a time!
7: 4y + 7(3x โ 8) 8: 5x + 6y(3x) โ 7y(-3)
9: 7 โ 3y(-4)(-7) + 4(5x) 10: 5x + 3(5y โ 6t) - 8
11: -(-3x)(-8) โ (5y โ 10) 12: -3t(2z) โ (4x - 8) + (-8y)
13: -(x)(-y)(-t) + cos(90) โ 3x(2t) 14: 3
4 5๐ฅ โ
2
3๐ฅ
4
7๐ฆ + 3๐ฅ(
5
6๐ก)
15: 3 2๐ฅ โ 3๐ฆ โ 7๐ฅ 2
5๐ฆ โ (โ๐ก) 16: โ 3๐ฅ โ ๐ฆ โ 17๐ง โ โ4๐ฅ
1
3 โ6๐ฆ +
4
5๐ฅ(โ9๐ก)
TBU: 3๐ฅ 5 โ 7๐ฆ + 1
3
2
7๐ฆ โ 6๐ก + 3๐ก โ7๐ฆ +
2
3๐ง 4๐ฅ โ
5
8๐ฆ + 8 โ
3
5๐ก ๐ฅ โ
1
2๐ง +
4
7 โ
3
8 (โ
3
4)
PEMD: The Bracket Way Page 17
The Bracket Method: there is a simple way to learn algebra 2009-2010
Algebra Multiplicationโฆhow good are you?
Part I: Bracket terms; donโt do the math; just label each term M, D or S.
1: 5x(-5) โ (-3)(-5) โ 7y(3x โ 10) 2: 4!(-7y) - .387(2x) โ 4cosx
3: 7 โ 4(3x โ 12) + 8 โ 7y 4: 6 28 + 5๐ฅ2(2๐ฆ3 โ 8) โ 2 (4)sin๐ฅ
Part II: Mash or Distribute each term
5: -(3x โ 5) 6: -(4x)(-5)
7: 2(4x)(+7)(-8) 8: -2(7 โ x)
Part III: Find the mistakes!
9: 5(3x) โ 7y โ (4t โ 8) 10: 6 + 4(4x โ 8y)
11: 5x โ 7x(3t) + 4 โ (2)(-x) 12: -(3x โ 7y) โ 3 + (3t โ 8z)
Part IV: Put it all together now: bracket, decide and label each term with (M or D or S), and do the math one term at a time.
13: 6(3x) + 5(3y โ 17) โ (-6)(-3v) 14: -(-6)(-5x) โ 7 + 3(4y โ 8z)
15: -5(2y) โ 9x(-7y)(-2) โ 8(-3) 16: 6(4 โ 7x) โ 2y โ (7t โ 9z) โ 3(-2x)(-z)(-t)
15x 7y -4t - 8 10 + 4x โ 8y
2x + 3t + 8x -3x -7y -9y +24z
Basic Exponents Page 18
The Bracket Method: there is a simple way to learn algebra 2009-2010
Basic Exponents Page 19
The Bracket Method: there is a simple way to learn algebra 2009-2010
What About Exponents?
Mash and write the answers in expanded notation (the long way)
1: 2x(3x)(-5x) 2: -4x(2y)(-3x) 3: -x(4x)(2t)(-5x) 4: -2x(3t)(4x)(-10t)(-t) Part b: Now, in the box below, rewrite your answer using exponents (exponential notation)
Distribute and write the answers in expanded notation
5: 3x(2x + 5y) 6: -2y(6x โ 5xy) 7: -xy(2x + 4y โ 2xy) 8) โx(x โ y + 2xy) Part b: Now below your expanded answer, write the answer using exponents
Put it all together: find terms, mash or distribute; use exponents if necessary
9: 2x(-3x) - (2y)(3y)(-4y) + 3x(2x)(-5x) 10: -3(2x)(-x) + 3(-5y)(-6y) โ (-2y)(3x)
11: 3x(2 โ 4x) + 3(2y)(-3y) โ x(xy) 12: 2xy(3y โ 4x) โ 5y(y)(-y)
13: -(2x)(-5x)(-y) + 5y(3x)(-2y) 14: 3(4x)(5x) โ 5y(2y)(-10y) + 6(2)(-10)
TBU: 2
3๐ฅ 4๐ฅ โ
5
4 3๐ฅ โ
5
6 + 4๐ฅ
7
8๐ฅ
8
9๐ฆ + 2๐ฆ โ
2
5๐ฆ + 4๐ฅ๐ฆ โ
4
3(6๐ฅ๐ฆ)(8๐ก)
Speak Geek
Exponent: math notation used to show repeat multiplication (mashing) of numbers or variables (ie. xxx=x3)
Expanded Notation: writing terms without the use of exponents (ie. xxxyy instead of x3y
2)
Alphabetical Order: it doesnโt really matter, but mathematicians do prefer you write variables in alphabetical order
Basic Exponents Page 20
The Bracket Method: there is a simple way to learn algebra 2009-2010
What About Exponents? Write the following expanded answers in exponential form
1: 7xx2x3 2: -42y3xy2x5 3: -x4x2txt3y2 4: -2xy3t4xt3t2y Math Geeks Only: What property allows you to re-arrange 5x3yxy2 into 5xx3yy2? (hint: change order?!)
Mash and write the answers in expanded notation (the long way)
1: 2x(3x)(-5x2) 2: -4x3(2y)(-3x2) 3: -x(4x3)(2t)(-5x) 4: -2x(3t2)(4x2)(-10t)(-t3) Part b: Now, below your expanded answer, write the answer using exponents
Distribute and write the answers in expanded notation
5: 3x(2x3 + 5y) 6: -3y2(6x โ 5xy) 7: -xy2(2x + 4y โ 2x3y) 8: โx(x2 โ y + 2x3y) Part b: Now below your expanded answer, write the answer using exponents
Put it all together: bracket, mash or distribute; use exponents if necessary
9: 2x(-3x) - (2y)(3y)(-4y) + 3x(2x)(-5x) 10: -3(2x)(-x) + 3x(-5y)(-6y) โ x(-2y)(3x)
11: 3x(2x โ 4y) + 3(2y)(-3y) โ x(2xy) 12: 2x(3 โ 4x) โ 5y(2y)(-y)
13: -(2x)(-5x)(-y) + 5(3y)(-2y) โ 3(2x)(-5) 14: 3(4x)(5x) โ 5y(2y)(-10y) + 6(2)(-10)
Basic Exponents Page 21
The Bracket Method: there is a simple way to learn algebra 2009-2010
The Challenge
The problems below are considered tough! If you can handle these, you โofficiallyโ have algebra skills!
1: 5x(3x) โ 4(x โ 7) โ (-8)(-y) 2: 4 โ 5(4x โ 8y) โ (x)(3x)(-x)
3: 5y(3x โ 4y) โ 4xy(5x) + 6y(-y)(2y) 4: 5x3(-3x2) โ 2x(5x2 โ 7y2) + y2(4y3)
5: -x(-x2)(-x3) + x2(y2) โ (x2 โ y2) 6: -2xy(4x2) + 2xy(3x โ 4y) โ 4x2(5x3)
7: 2x3(3x) โ 4y2(3y โ 4x) - 5x3 8: -(-3x2)(x4) โ 5(4x โ 7y) โ 6 + (3x)(-2x2)(3y)
9: 3x2y3(5x2 โ 7y3) + x3y(5xy2 โ 7xy) 10: 2x(-2y) โ 4x2(x2 - 5x) โ xy(-x3)(-y2)
11: 2
5๐ฅ โ15๐ฅ2 โ
1
3๐ฆ2 9๐ฆ โ 21๐ฆ3 12:
2
3๐ฅ 4๐ฅ
3
5๐ฅ2 โ
1
3๐ฆ2(
6
7โ
3
4๐ฆ3)
TBU: โ(โ3๐ฅ) โ5๐ฅ โ 3๐ฅ2 โ5๐ฅ4 +1
3๐ฅ 6๐ฅ2 โ 7๐ฆ3 +
5
6๐ฅ2๐ฆ3
9
10๐ฅ๐ฆ2 โ
3
4๐ฅ(
5
8๐ฅ3 โ 9๐ฆ)
Clean-up: Combine Like Terms Page 22
The Bracket Method: there is a simple way to learn algebra 2009-2010
Clean-up: Combine Like Terms Page 23
The Bracket Method: there is a simple way to learn algebra 2009-2010
Clean-up: Combine Like Terms Page 24
The Bracket Method: there is a simple way to learn algebra 2009-2010
Combine = โClean-upโ
Learn how to identify like terms: use underlines, double underlinesโฆ, and circle constants (regular numbers).
1: 5t โ 7y + 4 - 8t โ 12t + 8x 2: 3x + 6x2 -7xy - 18x2 + 10x โ 17xy
3: 4xy โ 7x โ xy2 + 8xy โ x + 3x2y 4: 3x โ 5y โ 3x2 โ 6x3 + x โ 7x4 + 51 - 51x3
5: 3 2 + 6 3 โ 2 2 + 2 โ 7 3 6: 5t โ 7t โ t2 + 10t3 โ 5 โ 6t2 + 9t3 โ 6
7: 6x โ 3x2 โ 10xy +8x2y โ 6xy2 + 6x2 โ 8x3 + x2 โ 7xy + 15x2y -3x2y2 โ 15 + 4xy โ x +2x3
Clean these up...combine the following expressions
8: 7x โ 12x + 8x โ 20x + 11x 9: 3y + 11y + 8y + 20y โ 40y
10: 3t2 + 9t2 - 4t2 โ 21t2 + 6t2 11: -8x5 + 9x5 โ 3x5 โ x5 +8x5
12: -101x + 56x โ 68x -71x -211x + 57x 13: 15y + 9y -12y + y โ 5y
Cleaning up (combining) is all about adding or subtracting; the sign tells you what to do.
14: 5x - 6x + 8x + 9y + 18y โ 7y 15: -9y + 7x โ 13x + 18y + 9x -21y โ 5x
16: 8x โ 5x2 โ 11x2 + 7x - 17x2 + 8x 17: 12x โ 15y + 9 + 4y โ 17 โ 9y + 10x - 5
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Coefficient: a number in front of a variable
Constant: a number by itself (no variables): a regular number (all constants/regular numbers are like terms)
Like Terms: terms where the non-coefficient parts (the โstuffโ after the front number) are the same
Clean-up: Combine Like Terms Page 25
The Bracket Method: there is a simple way to learn algebra 2009-2010
Combine = โClean-upโ
Clean-up the mess: Identify a term, find all of them, and combine (add or subtract depending on the sign)
1:
2:
3:
Answer:
Answer:
Answer:
Clean-up: Combine Like Terms Page 26
The Bracket Method: there is a simple way to learn algebra 2009-2010
Combine = โClean-upโ
Clean-up the mess: Identify a term, find all of them, and combine (add or subtract depending on the sign)
1:
2:
Now that you have seen the worst, try some basic problems: do the same thing!
3: -5x + 17y + 12 + 13xy - 3y โ 11x + 8xy - 28
4: 4x2 โ 5x3 + 9xy โ 10x3 + 13xy2 โ 2x2y + 25xy โ 14x2y โ xy2
5: 2 3 โ 8 2 + 6 3 โ 2 โ 3 3 + 15 2
Answer:
Answer:
Clean-up: Combine Like Terms Page 27
The Bracket Method: there is a simple way to learn algebra 2009-2010
Combining Like Terms (Clean-up)
Math Geeks Only: If xx = x2, then why is x+x not x2?
Find the Mistakes
1: 4x โ 5x2 + 8 โ 6x2 โ 11x + 3x2 - 7 2: 6x + 18y - 4 + 17y โ y + 8x - 9
-4x - 11x2 + 1 14x + 35y + 13
Identify and combine (clean-up)
3: 3x โ 5y +6x โ 18 +29y โ 84x -17 4: 5x2 โ 7x โ 8x2 -18x + 34x2 โ 17x2
5: 5x + x2 + 17x โ 4x2 + 10 โ x + 5x2 - 34 6: 16x + 14 xy โ 5y โ 8xy โ 7y2 + 12 โ 9y2 - xy
7: โ5 2 + 7 3 โ 8 3 + 11 2 โ 2 3 + 3 2 8: 6x3 โ 4x2 + 28x โ 15 + 7x3 + 14x2 + 8x - 18
9: x + xy โ y + 4xy โ 4y + x + xy โ 3x โ 6y 10: 3x2 โ 4xy2 + 7y2 โ 7x2y โ 9y2 โ 2 +17xy2 + 11
TBU1: 3
4๐ฅ โ
1
2๐ฅ2 โ 6 + 7๐ฅ โ
2
3๐ฅ2 + 15 TBU2: โ11 โ
2
5๐ฆ โ
1
4๐ฆ2 โ
1
6๐ฆ + ๐ฆ โ
1
3๐ฆ2 + 7 +
5
8๐ฆ2
Simplify: The Bracket Way Page 28
The Bracket Method: there is a simple way to learn algebra 2009-2010
Simplify: The Bracket Way Page 29
The Bracket Method: there is a simple way to learn algebra 2009-2010
Simplify = Bracket
Find the Mistakes
1: 5x โ 3(5 โ 8x) + 3(-4) 2: 7 + 3(6x โ 2) - 5x
5x - 15 + 24x - 12 10(6x โ 2) - 5
29x + 27 60x โ 20 - 5
60x - 25
Simplify: (1) Bracket terms, (2) Mash, Distribute (or Solo) each term, (3) Clean-up
3: 17 โ 5(3x โ 8) 4: 5x + 2(-4) โ 7x(-5) - 20
5: 20(-2x) โ 7(-8) + 2x โ 3(-7) 6: 5(2x - 8) โ 4(9x - 5)
7: -3(4) โ (3x โ 6) + 5 + 7(-4x) 8: 5x โ 7(-2x)(-5) โ 2(x โ 4)
9: 3(4x2 โ 5x -8) โ 2x2 โ 4(3x) - (-2) 10: -(-4)(-3y) + 3x -2(-4x -7y) + 5(-4y)
TBU: 7(3x)(-7) โ 4x โ 6(-3)(-1) - (4x โ 8y + 7) + 14y โ x(-5) โ 8(-3y + 8x) + 5 + 2(- 7)(-6) โ x โ y
Simplify: The Bracket Way Page 30
The Bracket Method: there is a simple way to learn algebra 2009-2010
The Challenge (Part II)
If you can simplify theseโฆyou can simplify anything!
Remember: (1) find terms, (2) Mash or Distribute each term, (3) Clean-up (combine)
1: 5x(2x) โ 5(3x2 โ 7x - 8) + (-7) 2: -2(-3x3) โ 2x(-5x) โ 6x2 + 5x(-2x)(-7x)
3: 2(x3 โ 18x2 + 9x โ 34) โ (x3 โ 8x2 โ 14x - 21) 4: 5(2x โ 6y) โ (4x)(-3y) โ 7(-2x) + x(3y) โ (-x)
5: x(x) โ 2(3x2 โ 7x โ 9) โ 3(2x) - 18 6: -(-6x3) + 18x2 โ 2x2(-5x) โ x(x โ 5)
7: 5(-2y) โ 7y(4y โ 8) โ 9 +2(7y) โ (-4y2) 8: 2xy(3x โ 8y) โ 3x(-4y2) โ x(9x)(-3y) + 7y(-2xy)
9: x(x3 โ 9x2 + 4x โ 21) โ 2x(x2 โ 4x) โ 2(-11) 10: 3
8๐ฅ 4๐ฅ โ 5๐ฅ2 +
2
3 ๐ฅ2 โ 3๐ฅ โ 8 โ
3
4(โ7)
TBU: -4x2(3x2) โ 7x3(-5) โ x(x3 โ 6x2 โ 10) โ x(-4x)(-7x) + 18x2(-x) + 5x3(-7x) โ 6x(18 โ 7x โ x2 + 5x3)
Solving: Junk & Divide Page 31
The Bracket Method: there is a simple way to learn algebra 2009-2010
Solving: Junk & Divide Page 32
The Bracket Method: there is a simple way to learn algebra 2009-2010
Solving: Junk & Divide Page 33
The Bracket Method: there is a simple way to learn algebra 2009-2010
Solving: Junk & Divide Page 34
The Bracket Method: there is a simple way to learn algebra 2009-2010
Solving Equations: Junk & Divide
Solving Basic Rules:
1. Draw the ___________ 2. Work on the _____________________________________ 3. Whatever you do to one side of the wall, _______________________________________________
Draw the wall; put an * above the junk (do not solve!)
1: 7x โ 27 = 76 2: 35 = -5 + 7x 3: 16 โ 8x = 123 4: -352 = 21x + 7
5: 4x - 7ฯ = 237 6: 71 = 2 3 โ 7๐ฅ 7: 9๐ฅ โ 4
7= 78 8:
6
5+ 7๐ฅ = 89
Now go back (1-8) and show how you would โget rid of the junkโ (do not dive yet, that is next)
Solve: Put it all together nowโฆJunk & Divide
9: 17 โ 9x = 71 10: -5x โ 35 = -72 11: 89 = 4x - 12
12: -57 = -9 + 17x 13: 12 โ 7x = -81 14: 13x + 78 = -114
15: 171 = 5 โ 5.2x 16: 17 โ x = 58 17: 5๐ฅ +7
3= 16
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Equation: when one mathematical expression is set equal to another.
Solving: Junk & Divide Page 35
The Bracket Method: there is a simple way to learn algebra 2009-2010
Solve: Junk & Divide
1: -17x + 167.4 = -431.5 2: -89 + 63.2x = 178.9 3: 54.5 = -76.2 โ 18.4x
4: 1235 = 123x โ 86 5: 1 โ x = 0 6: 67 โ 12x = -782
These are so easy, they are tough (sometimes you only do junk, and sometimes you skip right to divide)!?
7: 5x = 876 8: -15 + x = 91 9: -17 = -4x
10: -346 = 89 + x 11: - x = -86 12: x โ 13 = -21
Donโt panic, theyโre only fractions (remember to use parenthesis)โฆjunk and divide!
13: 5๐ฅ โ 2
3 = 28 14: 17 +
5
6๐ฅ = โ 34 15: 174 = 115 โ
2
5๐ฅ
16: 6 = 3
8๐ฅ โ
15
3 17:
5
9๐ฅ โ
27
3= 34 18:
46
7=
1
4๐ฅ +
21
5
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One Step Equations: equations which can be solved in one mathematical step.
Solving: Junk & Divide Page 36
The Bracket Method: there is a simple way to learn algebra 2009-2010
Simplify and Solve Bracket, Junk & Divide
1: 7 + 2(3x โ 8) = 123 2: 5x โ 2(3x) โ 18(-4) = 231
3: 4(-2) โ 6(3x โ 8) + 8(-3x) = -500 3: -5x โ 4(x โ 8) = -324
5: 1056 = 4(-2x) โ 18(3) โ 7 6: -78 = 7(2) โ 3(4x โ 8) + 2(-5x)
7: -(3x โ 8) โ x + 17 โ 3(-7x) = -378 8: 78 = 17 โ 2(x - 8) + 5(-7x) โ (-5)(-8)
TBU: (5x)(-17)(-4) โ (12x2 โ 6) โ 5x โ 4(5 โ x) โ (-2)(-8)(-3x) + 2(3x)(-7) + 4(3x2 โ 7x +9) = -1578
Solving: Junk & Divide Page 37
The Bracket Method: there is a simple way to learn algebra 2009-2010
Xโs on Both Sides โmake one disappearโ, junk & divide
1: 5x โ 19 = 54 + 8x 2: -15 + 6x = 25 + 8x 3: 24 โ 8x = -57 - 11x
4: 34 โ 18x = -71 + 9x 5: 8x + 19 = -6x โ 23 6: -16x + 31 = -7x + 23
7: 34.5 - 83.1x = 25.4x + 46.1 8: 5.26x + 8.12 = -8.95 + 9.74 9: -23 + 8x = 40 โ 8x
These are strangeโฆthink about which x you want to disappear!?
10: 13 + 9x = 6x 11: -78 โ 7x = -12x 12: 15x = 74 + 6x
13: -19x = -7x + 23 14: โ x = 75 + 8x 15: 40 + 9x = 19x
Hmmm?16: 6x โ 8 = 15 + 6x Hmmm?17: 21 โ 7x = -7x + 21
Solving: Junk & Divide Page 38
The Bracket Method: there is a simple way to learn algebra 2009-2010
The Challenge (Part III) If you can solve theseโฆyou can solve anything!
1: 6 + 2(3x โ 8) = 7x -9 2: -3(x โ 8) = -4(5x โ 7)
3: -4(-3) โ (6x โ 8) + 7 = 5x โ 18 4: 7x โ 8(-5x) + 7 = 17 โ 5x
5: 5(2x) โ 17(-2) = 2(7x โ 8) 6: -(-5) โ 4(2x โ 8) = 3x โ 7(-2) + 8x
7: 4x โ 2(-3x) + 2(5) = 7 โ 3(2x โ 8) โ 9x 8: 4(3x + 21) = 5x โ 7x + 15x
9: 5(-2)(3) โ (4x โ 9) โ 4(-7) = 3(-2) -3(4)(-5x) โ 9 10: 2(-4) โ x โ 2(3) = 8 + 8x โ 19 + 4x
11: 3
4 2๐ฅ โ 8 =
2
3 6๐ฅ โ 14(โ3) 12:
2
3 4 โ
4
5 3๐ฅ โ
3
8 =
1
2 4๐ฅ โ
7
8 โ2 (โ
2
21)
Solving: Junk & Divide Page 39
The Bracket Method: there is a simple way to learn algebra 2009-2010
Solving for Letters
When solving for formulas, nothing changes (except the answers are ugly looking). Identify what you are
trying to solve for; get rid of the junk; divide.
1: solve for x: 5x โ t = 4g 2: solve for t: 4g โ 7t = cy 3: solve for v: 3tv = 35g
4: solve for g: 16t2 โ 5g = F 5: Solve for r: 6ฯrh = 156 6: solve for h: fm โ gh = 38t
7: solve for v: 6t2 โ 7t โ 5vh = 324 8: solve for g: 7y โ 6t โ 8g = 56 9: solve for r: 7d โ 3ฯrh = -72g
One of the most common things you will do in algebra is to solve for y (get y alone!).
10: y + 4x = 7 11: 5x - y = 18 12: 5x + 8y = 21
13: 2x โ 7y = -21 14: 7y โ 4x โ 8 = 18 15: 15 = 5x โ 3y
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Implicit Formulas: solving a formula for another variable
Solving: Junk & Divide Page 40
The Bracket Method: there is a simple way to learn algebra 2009-2010
Adding to your repertoire
Junk & Divide can be used to solve most any type of linear algebraic equation. However, once you get good at
this method, there are certain methods you should learn to make solving faster or easier!
1) Cross Multiplication: used anytime you have fractions set equal to each other.
2) โDotโ Method, or IHF (I hate fractions), or GCF (this is the official โgeekโ name: it stands for greatest
common factor): used when you have a lot of fractions in an equation, and you just want to get rid of
them!
3) Graph the equation and find the intersect: every equation can be solved this way.
4) Two Sticks, Two Equations: used to solve absolute value problems.
5) The Inverse of Square is Square Root: Used to solve x2 or ๐ฅ problems