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Maelen's test > Algebra > Flashcards

Algebra Flashcards

1
Q

Simplify
5+(2x4+2)^2- |7-(-40|+18 / 3x5-8

SGAlg14

A
  1. Order of operations. PEMDAS: Parentheses-Exponents-(multiplication/division)-(addition/subtraction)

use Fraction bars as grouping symbols. numerator in parathesis and denom in paran

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2
Q

what are linear equations?SGAlg19

A

linear equations are equations in which all variables have an exponent of 1.

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3
Q

How do you simplify expressions? SGAlg19

A

1.Combine like terms
2.Find a common denominator
3. Pull out common factor
4. cancel common factors
value of expression stays the same.

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4
Q

How do you simplify equations? SGAlg19

A
  1. add the same thing to both sides
  2. subtract the same thing from both sides
  3. multiply both sides by the same thing
  4. divide both sides by the same thing
  5. raise both sides by the same power
  6. take the same root by both sides
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5
Q

identify equation and simplify
3x+5=26

SGAlg20

A

Solve one variable equation: isolate variable to one side.

x=7

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6
Q

solve the following for X and Y

x+y=9
2x=5y+4
SGAlg21

A

Method 1: Substitution
Solve for X then substitute in for x in second equation.
Method 2: Combination
line up the terms in the equation,
multiply one equation by some number: goal is to make either the coefficient of one of the variables the same in both equations, which you subtract one equation from another OR, you make the coefficient in front of one of the variables the same but with opposite signs, then you add the equations.
then solve for unknown

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7
Q

solve for w given that
12+ |w-4|=30

SGAlg21

A

Absolute value equations: each has two numbers that the variable can equal

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8
Q

If N employees are fulfilling orders at the rate of 3 orders per employee per hour, how many orders are filled in 4 hours?

SGAlgSN25

A

12n.
smart numbers allow to slove alg with arithmetic.
When to use: The unknown in the prob will never have a real value in the problem or answers. If an answer has a variable, or only refers to percents or ratios, use SN. Not an equation or inequality in answer. start with “2”
1. choose SNs to replace the unknowns.
2. Solve: add the SN back into the prob where ever it mentions the variable. and solve the math
3. Find a match in the answers by plugging in SN into the answers.

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9
Q

When do you choose smart numbers?

A

When the problem contains -only unspecified values: variables, only refers to percents, fractions or ratios.
-No real numbers in the answer choices
only variables, fractions or percents

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10
Q

Which smart numbers should you choose? SGAlgSN28

A

Dont pick 0 or 1
dont pick any number in the problem
choose numbers with different properties (odd and even), if you have to pick multiple numbers.
if you still get more than one answer as a match and have time, change only one number and do it again.
choose 2, 3, 5 to start with

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11
Q

What do exponents represent?

A

short hand for repeated multiplication

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12
Q

What is x?

x=x^2

A

x must be 1 or 0.

0 to any power= 0. 1 to any power=1

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13
Q

(3/4)^3 greater than 3/4?

A

as an exponent increases, the value of the positive proper fractions and decimals (between 0 and 1) decreases. since the denom is bigger and is being multiplied by itself, it is increasing faster than the numerator SGAlgEx35

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14
Q

what is the exponent rule used to simplify this expression: (3x)^4

A

(3x)^4=3^4 x x^4 = 81x^4 Compound Base: Exponents can be distributed to a fraction and a product (10^3)= (2x5)^3
SGAlgEx35

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15
Q

simplify:
(-2)^4
-2^4

A

-16, 16

negs outside of the parentheses does not distribute SGAlgEx35

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16
Q

simplify and identify rule:

m^6 x m^15

m^6/m^5

m^0

m^2/m^5

(x/4)^-2

(m^2)^5

A

Combining exponential terms with common bases:
m^21.

multiplying terms with same bases= add the exponents

m: dividing terms, subtract exponents
1: anything to the power of 0=1 bc anything divided by itself is 1.

m^-3 or 1/m^3
(4/x)^2
negative exponents: something with a negative exponent is just one over that same thing with a positive exponent.

m^10: nestled exponents: multiply exponents

SGAlgEx35

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17
Q

state if the value gets bigger or smaller:

(-3/2)^2
(-1/2)^2
(1/2)^2
(3/2)^2

(-3/2)^3
(-1/2)^3
(1/2)^3
(3/2)^3

A

any fraction less than 0 becomes bigger with a positive even exponent

any fraction more than 0 becomes bigger with a positive even exponent

any fraction less than -1 becomes smaller with a positive odd exponent

any fraction between -1 and 0 becomes bigger with a positive odd exponent

any fraction between 0 and 1 becomes smaller with a positive odd exponent

any fraction between greater than 1 becomes bigger with a positive odd exponent

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18
Q

solve for x

x^2=25

A

x= + or - 5. Even exponents hide the sign of the base. think of the bases as absolute values they can be either sign. Alg38
If an equation includes some variables with odd exponents and some variables with even exponents, it is dangerous-it likely has two solutions

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19
Q

solve for x

x^3=-125

A

x=-5 equations with neg exponents only have 1 solution
If an equation includes some variables with odd exponents and some variables with even exponents, it is dangerous-it likely has two solutions

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20
Q

solve for w

(4^w)^3=32^w-1

A

w=-5. be careful if the base is 0 or -1 or could be the base, because raising those powers does not change the value.

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21
Q

what is the value of sq root(16)? if x^2=16 what is x?

A
    • or -4
      if GMAT gives you a sq rt symbol only use the positive root. If it gives a squared variable and you take the square root, use both positive and neg solutions.
      odd roots only have one solution and keep the sign of the base.
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22
Q

Simplify 216^(1/3)

A
  1. the numerator tells you what power to raise the base to and the denominator tells you which root to take. You can raise the base to the power and take the root in either order.
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23
Q 
Solve:
sq rt(50) x sq rt(18)
sq rt(25x16)
A

break into factors to get squares and take the square root of each factor.
division is the same. cannot do with addition or subtraction of bases
sgalg45

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24
Q

simplify sq rt(52)

A 
sq rt (2x2x13)= 2sq rt(13)
split it out into prime factors
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25
Q

what does a quadratic equation look like?
And how do you factor it? Solve for the root (root equals solution=x).
x^2+3x+8=12

A

A quadratic equation is an exponent equation.
ax+bx+c=0 or another form. KEEP EYES OUT FOR ONES IN DISGUISE. if there is a variable squared and the same variable in another term, be wary. SGAL30

  1. Move all the factors to the left side and set to =0.
    x^2+3x-4=0
  2. Look at the constants. A=1, B=3 and C=-4. If A does not=1, divide each factor by A.
  3. In order to factor, find two integers who’s sum is 3 and product is -4. 4 and -1 work.
  4. write equation in factor form: (x+?)(x+?)
    (x+4)(x-1)
  5. Solve x for 0. X must equal -4 or 1.
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26
Q

Solve for w:

3w^2=6w

A 
0 or 2.
This is a disguised quadratic equation. 
3w^2-6w=0
w(3w-6)=0
w could equal 2 or 0
sgal51
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27
Q

solve for b:

(36/b)=b-5

A

b=9 or -4
1.multiply both sides by b and factor
sgal51

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28
Q

solve for x:

x^3+2x^2-3x=0

A

x=0, -3, and 1

  1. DO NOT divide each side by x. factor an x out from each term
  2. factor the quadratic

RULE: If you have a quadratic expression equal to 0 and you can factor an x out of the expression, then x=0 is a solution of the equation sgal51

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29
Q

solve:

(z+3)^2=25, what is z?

A

z=2, -8

sgal51

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30
Q

write as a quadratic

x+7)(x-3

A

x^2+4x-21. FOIL : multiply together First terms, outer terms, inner terms, last terms.

31
Q

How can you tell a perfect square quadratic?

A

It only has one solution
x^2+8x+16=0
(x+4)^2=0
x=-4

32
Q

solve for x:

(x^2+x-12)/(x-2) =0

A

x= 3 or -4. x cannot equal 0 because that would make the equation undefined

33
Q

What are the three special products?

A

x^2-y^2=(x+y)(x-y)
x^2+2xy+y^2=(x+y)(x+y)=(x+y)^2
x^2-2xy+y^2=(x-y)(x-y)=(x-y)^2

sglag52

34
Q

Factor: x^2-y^2

A

(x+y)(x-y)

35
Q

Factor: x^2+2xy+y^2

A

(x+y)(x+y)

(x+y)^2

36
Q

factor:

x^2-2xy+y^2

A

(x-y)(x-y)

(x-y)^2

37
Q

factor a^2-1

A

(a+1)(a-1)

38
Q

what is the value of x/y ?

  1. (x+y)/y=3
  2. y=4
A

A. DO NOT solve for each variable. Solve for the combo. Ask: how do I make the equation look like the answer?
split the fraction. (x/y)+(y/y)=3

39
Q

if x does not =y, what is the value of x+y?

  1. x-y=1
  2. x^2-y^2=x-y
A

B. Combo. do not solve for each variable, solve for the combo.
RULE: if you are given just one linear equation with only basic math operations (add, subtraction, multiplication, or division) (like #1), you cannot alter the initial relationship between the two variables.

40
Q

If a=3bc and abc does not =0. what is the value of c?

  1. a=10-b
  2. 3a=4b
A

B. rephrase: what is a/b?
This is a combo problem in disguise. if the stem gives information in addition to a question of a single variable, see how the info can be combined.
sgalg60

41
Q

What are the four major types of formulas on the gmat and what do they look like?

A
  1. Plugin formula, problem gives you values for certain variables and asks you to solve for one variable.
  2. function. think “magic box”, a function is dependent on the value of an independent variable. ex: f(x)=4x^2-11. the value of the function f (output or range) is dependent on the value of x (input or domain). The domain can also be an algebraic equation
  3. Strange symbol formula: the symbol represents a procedure. Say the procedure aloud naming the first and second numbers. eg: x @ y= x^2+y^2-xy: First number squared plus second number squared.. Formulas that act on decimals [5.1] are included
  4. Sequence formulas: a sequence is a collection of numbers in a set order. every sequence is defined by a rule. A(subn)=9n+3
42
Q

if a (subn)=2a(subn-1)-4 and a (sub6)=-4, what is the value of a(sub4)?

A
  1. Recursive sequence defines each term relative to other terms. sgalg70
43
Q

if each number in a sequence is 3 more than the previous number and the 6th number is 32, what is the 100th number?

A
  1. Use reasoning instead of trying to find the rule. sgalg70
44
Q

You can perform all operations on inequalities the same as equations except which operations?

A

Multiplying or dividing by a negative number. When you do, flip the sign. dont div or mul variables in inequalities unless you know its sign. asvalg9,75

45
Q

is a+2b

A

C. Combine inequalities by adding them up and to get it to look like the original inequality. in this example, you have to add the second inequality twice.
never subtract inequalities or divide. for multiplying, make sure the possible values of the inequality is positive

46
Q

if 10+x^2=9 what is the range of possible values of x

A

if x is positive it is greater or equal to 3, if x is negative than it is less than or equal to -3. because when you square root a squared, remember there are two values

47
Q

Why, when and how should you use smart numbers?

A

Why: Arithmetic is always easier than algebra. Algebra takes a lot of time & energy

When: There are unknown values in the question. When the AC’s have variables, percentages or fractions, the question is a good candidate for SN.

How:

  1. List your unknowns ask “What do I want to know that is not given?”
  2. Think about which unknown you should pick a SN for. This should be the variable that doesn’t depend on the rest.
  3. Put the unknowns into the answer choices to see if you get the result when you solved using the SN. If it takes too long to solve, try to eyeball the result instead of solving the AC all the way.

If two AC’s have same outcome, then guess by looking at the AC. or try a different SN fast!

48
Q

What is the difference between expressions and equations?

A

In Expressions, there isn’t an equal sign and values must remain the same. With an Equation- you can alter both sides of the equal sign and values do not need to remain the same.

49
Q

How do you eliminate a variable when you have two equations?

A

You can plug in or you can stack the equations. When you stack, make one of the variable coefficients equal to the same variable in the other equation and then subtract the equation.

50
Q

How do you know when to work backwards?

A

Sometimes you dont. Start with alg, but once you realize it’s hard, start plugging in answer choices. Start with B and D.
A few signals:
1. Answer choices are easy numbers (small integers)
2. It’s problem solving and not DS
3. You are solving for only one value.

KEEP YOUR ALGEBRA AND WORK CLEAN MAELEN!!!

51
Q

What is the high school rule and what is it good for?

A

That when you have 2 unknown variables you need 2 equations to solve. This is good to use in PS but not DS because DS can actually be asking for combos or ratios which you dont need 2 equations for.

52
Q

When you are straight struggling with algebra what should you do?

A

Ask yourself, “What are the things I want to know but dont?” Then write them out and see how you could solve for the most important.
PS114

53
Q

What should you do if you dont know what algebraic equation to write?

A

Think logically and use real numbers. Pretend the variables are real numbers and ask yourself “what would happen?” to get the result.

54
Q

x^2-y^2

A

(x+y)(x-y)

This one is the trickiest as it can hide in a variety of formats like 1-n^4

55
Q

x^2+2xy+y^2

A

=(x+y)(x+y)=(x+y)^2

56
Q

x^2-2xy+y^2=

A

(x-y)(x-y)=(x-y)^2

57
Q

What is the equation for compound interest?

A

Total= P(1+ (r/n))^nt

P=principal
R=rate in decimal
n=number of times per year
t=number of years

58
Q

What should you think immediately when you see xy>0

A

X &Y have the same signs. both are either positive or negative.

59
Q

What should you think immediately when you see xy

A

X & Y have different signs. One is positive and one is negative.

60
Q

How should you translate X^2 - X

A

X^2

61
Q

What’s your strategy for approaching inequality problems?

A

Try algebra first and manipulate the inequality. Plan B is to test cases.

62
Q

What is your strategy for approaching system of inequalities?

A

Line them up with signs aligned and facing the same direction! Do not subtract or divide inequalities.

63
Q

What should you expect when solving quadratic equations?

A

2 answers

64
Q

What should you do if you have a number like 399 appear when you’re simplifying quadratics?

A

It’s 1 less than a perfect square. So, you probably missed an important step that would make it easier to solve

65
Q

Simplify 3w^2=6w

A

Watch for disguised quadratics when finding solutions. This is a quadratic and has two solutions

3w^2 -6w=0
w(3w-6)=0
w=2 or 0

66
Q

What is the plugin formula? And the approach you should take?

A

When you need to plug in variables and numbers into the equation. Plug in the numbers and remember to simplify and solve for the unknown that the question asks about.

67
Q

What is a function? and what is the approach you should take?

A

Functions look like F(x), think of it like a magic box. The DOMAIN is what you put into the box. the RANGE is what comes out of the box.

68
Q

What is a strange symbol formula and what is the approach?

A

A formula where the GMAT introduces an arbitrary symbol and uses it to define a procedure. Read it in relation to numbers. X$Y=X^2+Y^2-xy “First number squared plus second number squared…”

69
Q

What are sequence formulas?

A

A collection of numbers in a set order.

A (sub nth)=9n+3

70
Q

What is a recursive sequence?

A

Defines each term relative to another term

a(of n)=2a(of n-1) -4 if a(6th)=-4 then what does a(4th)=?

71
Q

If each number is 3 more than the previous and X( 6th term)=32, what is the 100th number?

A

100 terms-6terms=94 terms
each term is 3 more than the previous so 3x94 terms =282
94terms +6 terms= 282+32=314.

72
Q

When do you flip the sign of an inequality?

A

When you divide or multiply by a negative number. DO NOT divide or multiply with a variable that you don’t know the sign for.

73
Q

How do you combine inequalities?

A

Line them up. Simplify individual inequalities and have signs face the same direction.
Never subtract or divide 2 inequalities. You can multiply if all are positive but that is rare.

Maelen's test (21 decks)

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