bracket algebra

No More Double Signs

The Bracket Method: there is a simple way to learn algebra 2009-2010



No more double signs

A “+”, or “-” sign can mean many things; create a list of words for both signs:

“+”: positive, plus, add, _________________________________________________________________

_____________________________________________________________________________________

“-”: 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 – 46 + 71 + -26 2: 15 + 62 - +18 - -12 - 6

What happens when the first number is “-“? You must tell the calculator the starting point is negative! The only

time you use the “negative sign” when adding or subtracting is to tell the calculator the first number is negative.

3: -32 - +65 – 90 - -61 + 17 4: -28 – 18 + 92 + -28 + -65

5: -120 – 85 + -62 - +24 - 82 6: 18 – 9 + -68 + 25 – 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




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 “5 + -10” or “6 - +7”. A number is either + or –!

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 tougher…remember 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 – signs (no equal sign).



No more double signs

Find the Mistakes:

1: 51 + -36 – 46 + 71 + -26 2: 15 + 62 - +18 - -12 – 6 3: 18 - 52+ -53 - -12 – 97 + -31

51 - 36 - 46 – 71 – 26 15 + 62 – 18 – 12 – 6 18 – 52 + 53 + 12 – 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 – 90 + 61 + 17 5a: -120 + 85 - 62 + 24 – 82 6a: -14 + 75 - 12 - 34 + 45

4b: 61 – 32 + 17 – 65 – 90 5b: 85 – 62 - 82 + 24 – 120 6b: 45 – 12 - 34 + 75 - 14

Now that you know the Commutative Property works: find the mistakes; why don’t these problems work?

7a: -120 – 85 + 62 - 24 + 82 8a: -14 + 75 - 12 - 34 + 45 9a: -15 – 5 + 82 - 24 + 59

7b: -85 + 62 + 82 + 24 -120 8b: 45 – 12 + 34 – 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

9−

−1

4 +

4

7+

−6

7

11: −3

5 −

+1

8−

−3

4 +

−5

6+

3

7

TBU: 3

5+

−7

9−

+1

4 +

4

7−

−6

7

TI-83 A fraction is a division problem in disguise. To enter a fraction, simply type the top number, “÷”, and the bottom number. To be safe, put all fractions inside (); do not put the operation sign inside the ()

example: (3÷5) – (5÷7)

Don’t 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: “Math, one, enter”.

Speak Geek

Commutative Property: Do you see the “Co” in the word Commutative? Remember Co -“change order”. 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


How to Bracket


How to Bracket

Bracketing Terms: the first step to the bracket method is to be able to “bracket the terms” in an expression

What is the rule for bracketing:_____________________________________________________________

______________________________________________________________________________________

Bracket the following expressions

1: 5x2 – 6(3x) + 5y(8x) - 17 2: -6x(4y) – 7x(2y-7t) + 6z – 8(5) + 19(-t)

3: -2(5x) + 6(2x-7+8y) + 9(3)(-7) 4: 6x2(4x) – 2(5y)(-6)(-9y) – 6t – 7t3 + 2

5: -5 + 6y – (5)(-8x)(6y) - 8t(-6x)(3y4) – 4(-7x) – (-x)

Don’t panic, stick with your rules!

6: 5 6 + 7 − 3𝑥 − 8 7𝑥 − (2!)(6𝑥 − 5𝑦) 7: 5𝑥 7𝑦 − 6𝑥2 + sin 6𝑥 − cos(3𝑥 − 9)

TBU: -6(5x) + 6x(6y)(-6) – (4x +5-7y) – 8(4)(-6) + 5x – 6(-x)(-y) + 4 – 6(-8) – 2 + 9x(-3x)(-6x)

How well have you trained your eyes? Without bracketing, write the number terms in each expression?

8: 5x(-40) – 7(-3) – (3x-4) – (3-6y) 9: 4(3x)(4y)(-5z) – 2(3) – (53y) – 2(-5)

10: -6(3x)(-6y) – 6x – 6(4x-7) - 6(7) 11: -5x(7y)(-6) – 2x2 + 5(-3x-2y2) – x(5y) + (-t)

Speak Geek

Term: a part of a mathematical expression or equation separated by + or – signs.

How to Mash


How to Mash


Multiplication = Mash potatoes

In algebra, the easiest operation is multiplication. Just put it all the signs, numbers, letters in one pile (like your

uncle’s mash potato plate at thanksgiving: potato’s, 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)

Don’t forget about signs? (In multiplication and division, every 2 negative signs is equal to a positive!)

Even # of – signs = _____ Odd # of – 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 7𝑥 −

3

4

2

8𝑦 −5𝑡 −𝑡 (−8𝑧)

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? (ahem…change order)

How to Mash


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) – 5(-2x) + 7(3t)(-5x)

3: 3(-5) + 3x(-7y) - 3(-x)(y)(-t) 4: -(4)(-5) – 5(-2x)(-7) + 4(-6x)(y)

5: -(-4x) + (-6)(-5t)(7z) – x(-y)(-z) 6: 7(3) – 4(-8x) + 4x(2y) – 6(-8t)

7: x – 4t(6)(3) – 6 + 2x(-6) 8: -6(-4)(-5)(7) – (2x)(3y)(-6y)3

9: -(-1)(-1)(-5) + 3x(-3z) + 5(-y) 10: 3.2(-6.5) + 4x(-5.7)(-2.1) – 3.6z(-4.5t)

11: -12(7x)(-15y) – 5(-19)(6t) – (-12x)(-25) 12: -t – (-x)(-t)(-y) – (v)(-z)(-t) – x(z)

TBU: 3

5𝑥 7𝑦 − 5

2

3𝑥 −

4

7𝑡 + 5.6𝑡 −5𝑥

2

5𝑦 − −7.2𝑡 + 5.8

7

9𝑦

6

5𝑡 − 𝑥 −𝑦 −8 +

9

3

How to Mash


Mixed Review

Part I: No more double signs – you know what to do!

1: -32 - +65 – 90 - -61 + 17 2: -28 – 18 + 92 + -28 + -65

3: -120 – 85 + -62 - +24 - 82 4: 18 – 9 + -68 + 25 – 98 - -8 - +89

Part II: Bracket the following terms – don’t do the math, just see how many terms there are!

5: 5(2x) – 7(-3) + x(y)(-z) 6: sin32 – 14(x) + 4cos(3x) – 2(3)(4)

7: 3

4 4x-7 + 3 49-5x +

2

5 x (y) 8: 4 – (7 -2x) – (x)(2y) + 17(4) – 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 5𝑦

2

3𝑡 (−

3

8𝑥) 14: -(-x)(-y)(-t)(-z)

Part IV: Bracket terms; mash each term.

15: 2(-4) + 4x(-6y) - 3(-x)(5)(-t) 16: -(3)(-5) – 2(-3x)(-5) + 3(-6x)(y)

17: -(-4x) + (-3)(-2t)(7z) – x(-2)(-z) 18: 5(4) – (-8z) + 4x(2y) – 3(-7t)

How to Distribute (Pizza Delivery)




Distribute: the second type of multiplication

Write M (mash) or D (distribute) above each term…then do it!

1: 6(3x)(-5) 2: 7(5x + 7)

3: 7(x + 9) 4: -5(6 – x + 7y)

5: -(6x)(+7) 6: -(7 – 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 – 8x) 12: 2y( -7 – 4x)

13: - ( 5x – 8) 14: 5(t – x)

15: -5(2x – y + 5) 16: (6 – 5x + 7y)

TBU1: -2xy(6t – 9 + 4z – u) TBU2: 3

4(5𝑥 −

1

3𝑦 +

5

9𝑧)



Mixed Review

Part I: No more double signs – you know what to do!

1: -22 - +61 – 23 - -16 + 37 2: -18 – 38 + 52 + -8 + -25

3: -12 – 65 + -75 - +44 - 62 4: 8 – 19 + -38 + 27 – 81 - -15 - +64

Part II: Bracket the following terms – don’t do the math, just see how many terms there are!

5: 3(-2x) – 5(-2) + x(3y)(-4) 6: 32(7!) – (2x) + tan(3x) – 2!(3)(4)

7: -3 49 +3

4 4x + 𝑥2 4(𝑦)3 8: – (y -2x) – (-x2)(-18x)3 - (5x)(3y) + 17(4) – 2x3(5x)

Part III: Bracket terms; mash each term

9: 2(5x) - 4(-x)(2y) - 8(5y) 10: – 6(-2x) - 5x(4y)(2) + (4t)(-7x)

11: -(-7) - 2(-x)(y)(-7) + 3x(-7y) 12: -(3)(-2)(7)(v) – 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 – 5x – 8)

16: 2

3(

1

2𝑥 −

1

3) 17: -(-4x)(-7)(-6y) 18: -5(6 – x + 7y)

PEMD: The Bracket Way




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) – 3(2 +6y) 2: -2(5y) + 4(2)(5) – 3(2x)

3: -3(4 -5x) – 2(4y) + 2(3t)(-z) 4: 5(3x-6y) – 4(-5)(-3) – 4(t)

5: -(4x)(-5) – (5y – 7) 6: -3(2x) – 4(5 – 3y) – 4w + 2(3xy – 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 – 8) 8: 5x + 6y(3x) – 7y(-3)

9: 7 – 3y(-4)(-7) + 4(5x) 10: 5x + 3(5y – 6t) - 8

11: -(-3x)(-8) – (5y – 10) 12: -3t(2z) – (4x - 8) + (-8y)

13: -(x)(-y)(-t) + cos(90) – 3x(2t) 14: 3

4 5𝑥 −

2

3𝑥

4

7𝑦 + 3𝑥(

5

6𝑡)

15: 3 2𝑥 − 3𝑦 − 7𝑥 2

5𝑦 − (−𝑡) 16: − 3𝑥 − 𝑦 − 17𝑧 − −4𝑥

1

3 −6𝑦 +

4

5𝑥(−9𝑡)

TBU: 3𝑥 5 − 7𝑦 + 1

3

2

7𝑦 − 6𝑡 + 3𝑡 −7𝑦 +

2

3𝑧 4𝑥 −

5

8𝑦 + 8 −

3

5𝑡 𝑥 −

1

2𝑧 +

4

7 −

3

8 (−

3

4)



Algebra Multiplication…how good are you?

Part I: Bracket terms; don’t do the math; just label each term M, D or S.

1: 5x(-5) – (-3)(-5) – 7y(3x – 10) 2: 4!(-7y) - .387(2x) – 4cosx

3: 7 – 4(3x – 12) + 8 – 7y 4: 6 28 + 5𝑥2(2𝑦3 − 8) − 2 (4)sin𝑥

Part II: Mash or Distribute each term

5: -(3x – 5) 6: -(4x)(-5)

7: 2(4x)(+7)(-8) 8: -2(7 – x)

Part III: Find the mistakes!

9: 5(3x) – 7y – (4t – 8) 10: 6 + 4(4x – 8y)

11: 5x – 7x(3t) + 4 – (2)(-x) 12: -(3x – 7y) – 3 + (3t – 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 – 17) – (-6)(-3v) 14: -(-6)(-5x) – 7 + 3(4y – 8z)

15: -5(2y) – 9x(-7y)(-2) – 8(-3) 16: 6(4 – 7x) – 2y – (7t – 9z) – 3(-2x)(-z)(-t)

15x 7y -4t - 8 10 + 4x – 8y

2x + 3t + 8x -3x -7y -9y +24z

Basic Exponents


Basic Exponents


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 – 5xy) 7: -xy(2x + 4y – 2xy) 8) –x(x – 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) – (-2y)(3x)

11: 3x(2 – 4x) + 3(2y)(-3y) – x(xy) 12: 2xy(3y – 4x) – 5y(y)(-y)

13: -(2x)(-5x)(-y) + 5y(3x)(-2y) 14: 3(4x)(5x) – 5y(2y)(-10y) + 6(2)(-10)

TBU: 2

3𝑥 4𝑥 −

5

4 3𝑥 −

5

6 + 4𝑥

7

8𝑥

8

9𝑦 + 2𝑦 −

2

5𝑦 + 4𝑥𝑦 −

4

3(6𝑥𝑦)(8𝑡)

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 doesn’t really matter, but mathematicians do prefer you write variables in alphabetical order

Basic Exponents


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 – 5xy) 7: -xy2(2x + 4y – 2x3y) 8: –x(x2 – 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) – x(-2y)(3x)

11: 3x(2x – 4y) + 3(2y)(-3y) – x(2xy) 12: 2x(3 – 4x) – 5y(2y)(-y)

13: -(2x)(-5x)(-y) + 5(3y)(-2y) – 3(2x)(-5) 14: 3(4x)(5x) – 5y(2y)(-10y) + 6(2)(-10)

Basic Exponents


The Challenge

The problems below are considered tough! If you can handle these, you “officially” have algebra skills!

1: 5x(3x) – 4(x – 7) – (-8)(-y) 2: 4 – 5(4x – 8y) – (x)(3x)(-x)

3: 5y(3x – 4y) – 4xy(5x) + 6y(-y)(2y) 4: 5x3(-3x2) – 2x(5x2 – 7y2) + y2(4y3)

5: -x(-x2)(-x3) + x2(y2) – (x2 – y2) 6: -2xy(4x2) + 2xy(3x – 4y) – 4x2(5x3)

7: 2x3(3x) – 4y2(3y – 4x) - 5x3 8: -(-3x2)(x4) – 5(4x – 7y) – 6 + (3x)(-2x2)(3y)

9: 3x2y3(5x2 – 7y3) + x3y(5xy2 – 7xy) 10: 2x(-2y) – 4x2(x2 - 5x) – xy(-x3)(-y2)

11: 2

5𝑥 −15𝑥2 −

1

3𝑦2 9𝑦 − 21𝑦3 12:

2

3𝑥 4𝑥

3

5𝑥2 −

1

3𝑦2(

6

7−

3

4𝑦3)

TBU: −(−3𝑥) −5𝑥 − 3𝑥2 −5𝑥4 +1

3𝑥 6𝑥2 − 7𝑦3 +

5

6𝑥2𝑦3

9

10𝑥𝑦2 −

3

4𝑥(

5

8𝑥3 − 9𝑦)

Clean-up: Combine Like Terms




Combine = “Clean-up”

Learn how to identify like terms: use underlines, double underlines…, and circle constants (regular numbers).

1: 5t – 7y + 4 - 8t – 12t + 8x 2: 3x + 6x2 -7xy - 18x2 + 10x – 17xy

3: 4xy – 7x – xy2 + 8xy – x + 3x2y 4: 3x – 5y – 3x2 – 6x3 + x – 7x4 + 51 - 51x3

5: 3 2 + 6 3 − 2 2 + 2 − 7 3 6: 5t – 7t – t2 + 10t3 – 5 – 6t2 + 9t3 – 6

7: 6x – 3x2 – 10xy +8x2y – 6xy2 + 6x2 – 8x3 + x2 – 7xy + 15x2y -3x2y2 – 15 + 4xy – x +2x3

Clean these up...combine the following expressions

8: 7x – 12x + 8x – 20x + 11x 9: 3y + 11y + 8y + 20y – 40y

10: 3t2 + 9t2 - 4t2 – 21t2 + 6t2 11: -8x5 + 9x5 – 3x5 – x5 +8x5

12: -101x + 56x – 68x -71x -211x + 57x 13: 15y + 9y -12y + y – 5y

Cleaning up (combining) is all about adding or subtracting; the sign tells you what to do.

14: 5x - 6x + 8x + 9y + 18y – 7y 15: -9y + 7x – 13x + 18y + 9x -21y – 5x

16: 8x – 5x2 – 11x2 + 7x - 17x2 + 8x 17: 12x – 15y + 9 + 4y – 17 – 9y + 10x - 5

Speak Geek

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 “stuff” after the front number) are the same




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 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 – 11x + 8xy - 28

4: 4x2 – 5x3 + 9xy – 10x3 + 13xy2 – 2x2y + 25xy – 14x2y – xy2

5: 2 3 − 8 2 + 6 3 − 2 − 3 3 + 15 2

Answer:

Answer:



Combining Like Terms (Clean-up)

Math Geeks Only: If xx = x2, then why is x+x not x2?

Find the Mistakes

1: 4x – 5x2 + 8 – 6x2 – 11x + 3x2 - 7 2: 6x + 18y - 4 + 17y – y + 8x - 9

-4x - 11x2 + 1 14x + 35y + 13

Identify and combine (clean-up)

3: 3x – 5y +6x – 18 +29y – 84x -17 4: 5x2 – 7x – 8x2 -18x + 34x2 – 17x2

5: 5x + x2 + 17x – 4x2 + 10 – x + 5x2 - 34 6: 16x + 14 xy – 5y – 8xy – 7y2 + 12 – 9y2 - xy

7: −5 2 + 7 3 − 8 3 + 11 2 − 2 3 + 3 2 8: 6x3 – 4x2 + 28x – 15 + 7x3 + 14x2 + 8x - 18

9: x + xy – y + 4xy – 4y + x + xy – 3x – 6y 10: 3x2 – 4xy2 + 7y2 – 7x2y – 9y2 – 2 +17xy2 + 11

TBU1: 3

4𝑥 −

1

2𝑥2 − 6 + 7𝑥 −

2

3𝑥2 + 15 TBU2: −11 −

2

5𝑦 −

1

4𝑦2 −

1

6𝑦 + 𝑦 −

1

3𝑦2 + 7 +

5

8𝑦2

Simplify: The Bracket Way




Simplify = Bracket

Find the Mistakes

1: 5x – 3(5 – 8x) + 3(-4) 2: 7 + 3(6x – 2) - 5x

5x - 15 + 24x - 12 10(6x – 2) - 5

29x + 27 60x – 20 - 5

60x - 25

Simplify: (1) Bracket terms, (2) Mash, Distribute (or Solo) each term, (3) Clean-up

3: 17 – 5(3x – 8) 4: 5x + 2(-4) – 7x(-5) - 20

5: 20(-2x) – 7(-8) + 2x – 3(-7) 6: 5(2x - 8) – 4(9x - 5)

7: -3(4) – (3x – 6) + 5 + 7(-4x) 8: 5x – 7(-2x)(-5) – 2(x – 4)

9: 3(4x2 – 5x -8) – 2x2 – 4(3x) - (-2) 10: -(-4)(-3y) + 3x -2(-4x -7y) + 5(-4y)

TBU: 7(3x)(-7) – 4x – 6(-3)(-1) - (4x – 8y + 7) + 14y – x(-5) – 8(-3y + 8x) + 5 + 2(- 7)(-6) – x – y



The Challenge (Part II)

If you can simplify these…you can simplify anything!

Remember: (1) find terms, (2) Mash or Distribute each term, (3) Clean-up (combine)

1: 5x(2x) – 5(3x2 – 7x - 8) + (-7) 2: -2(-3x3) – 2x(-5x) – 6x2 + 5x(-2x)(-7x)

3: 2(x3 – 18x2 + 9x – 34) – (x3 – 8x2 – 14x - 21) 4: 5(2x – 6y) – (4x)(-3y) – 7(-2x) + x(3y) – (-x)

5: x(x) – 2(3x2 – 7x – 9) – 3(2x) - 18 6: -(-6x3) + 18x2 – 2x2(-5x) – x(x – 5)

7: 5(-2y) – 7y(4y – 8) – 9 +2(7y) – (-4y2) 8: 2xy(3x – 8y) – 3x(-4y2) – x(9x)(-3y) + 7y(-2xy)

9: x(x3 – 9x2 + 4x – 21) – 2x(x2 – 4x) – 2(-11) 10: 3

8𝑥 4𝑥 − 5𝑥2 +

2

3 𝑥2 − 3𝑥 − 8 −

3

4(−7)

TBU: -4x2(3x2) – 7x3(-5) – x(x3 – 6x2 – 10) – x(-4x)(-7x) + 18x2(-x) + 5x3(-7x) – 6x(18 – 7x – x2 + 5x3)

Solving: Junk & Divide




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 – 27 = 76 2: 35 = -5 + 7x 3: 16 – 8x = 123 4: -352 = 21x + 7

5: 4x - 7π = 237 6: 71 = 2 3 − 7𝑥 7: 9𝑥 − 4

7= 78 8:

6

5+ 7𝑥 = 89

Now go back (1-8) and show how you would “get rid of the junk” (do not dive yet, that is next)

Solve: Put it all together now…Junk & Divide

9: 17 – 9x = 71 10: -5x – 35 = -72 11: 89 = 4x - 12

12: -57 = -9 + 17x 13: 12 – 7x = -81 14: 13x + 78 = -114

15: 171 = 5 – 5.2x 16: 17 – x = 58 17: 5𝑥 +7

3= 16

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Equation: when one mathematical expression is set equal to another.



Solve: Junk & Divide

1: -17x + 167.4 = -431.5 2: -89 + 63.2x = 178.9 3: 54.5 = -76.2 – 18.4x

4: 1235 = 123x – 86 5: 1 – x = 0 6: 67 – 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 – 13 = -21

Don’t panic, they’re only fractions (remember to use parenthesis)…junk and divide!

13: 5𝑥 − 2

3 = 28 14: 17 +

5

6𝑥 = − 34 15: 174 = 115 −

2

5𝑥

16: 6 = 3

8𝑥 −

15

3 17:

5

9𝑥 −

27

3= 34 18:

46

7=

1

4𝑥 +

21

5

Speak Geek

One Step Equations: equations which can be solved in one mathematical step.



Simplify and Solve Bracket, Junk & Divide

1: 7 + 2(3x – 8) = 123 2: 5x – 2(3x) – 18(-4) = 231

3: 4(-2) – 6(3x – 8) + 8(-3x) = -500 3: -5x – 4(x – 8) = -324

5: 1056 = 4(-2x) – 18(3) – 7 6: -78 = 7(2) – 3(4x – 8) + 2(-5x)

7: -(3x – 8) – x + 17 – 3(-7x) = -378 8: 78 = 17 – 2(x - 8) + 5(-7x) – (-5)(-8)

TBU: (5x)(-17)(-4) – (12x2 – 6) – 5x – 4(5 – x) – (-2)(-8)(-3x) + 2(3x)(-7) + 4(3x2 – 7x +9) = -1578



X’s on Both Sides “make one disappear”, junk & divide

1: 5x – 19 = 54 + 8x 2: -15 + 6x = 25 + 8x 3: 24 – 8x = -57 - 11x

4: 34 – 18x = -71 + 9x 5: 8x + 19 = -6x – 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 – 8x

These are strange…think about which x you want to disappear!?

10: 13 + 9x = 6x 11: -78 – 7x = -12x 12: 15x = 74 + 6x

13: -19x = -7x + 23 14: – x = 75 + 8x 15: 40 + 9x = 19x

Hmmm?16: 6x – 8 = 15 + 6x Hmmm?17: 21 – 7x = -7x + 21



The Challenge (Part III) If you can solve these…you can solve anything!

1: 6 + 2(3x – 8) = 7x -9 2: -3(x – 8) = -4(5x – 7)

3: -4(-3) – (6x – 8) + 7 = 5x – 18 4: 7x – 8(-5x) + 7 = 17 – 5x

5: 5(2x) – 17(-2) = 2(7x – 8) 6: -(-5) – 4(2x – 8) = 3x – 7(-2) + 8x

7: 4x – 2(-3x) + 2(5) = 7 – 3(2x – 8) – 9x 8: 4(3x + 21) = 5x – 7x + 15x

9: 5(-2)(3) – (4x – 9) – 4(-7) = 3(-2) -3(4)(-5x) – 9 10: 2(-4) – x – 2(3) = 8 + 8x – 19 + 4x

11: 3

4 2𝑥 − 8 =

2

3 6𝑥 − 14(−3) 12:

2

3 4 −

4

5 3𝑥 −

3

8 =

1

2 4𝑥 −

7

8 −2 (−

2

21)



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 – t = 4g 2: solve for t: 4g – 7t = cy 3: solve for v: 3tv = 35g

4: solve for g: 16t2 – 5g = F 5: Solve for r: 6πrh = 156 6: solve for h: fm – gh = 38t

7: solve for v: 6t2 – 7t – 5vh = 324 8: solve for g: 7y – 6t – 8g = 56 9: solve for r: 7d – 3πrh = -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 – 7y = -21 14: 7y – 4x – 8 = 18 15: 15 = 5x – 3y

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Implicit Formulas: solving a formula for another variable



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) “Dot” Method, or IHF (I hate fractions), or GCF (this is the official “geek” 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 𝑥 problems

bracket algebra

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