Algebra Flashcards
To solve normally need as many equations as there are variables.
What are the exceptions?
When there are defined constraints
ie x is a positive integer ; y is a prime #
When there are inequalities
ie x < 5
x x
a * b
a+b
x
a b
x
ab
x
a b
x / x
a-b
x
-a
x
1
____
a
x
1
_____
-a
x
a
x
All positive numbers have how many square roots?
Two one positive and one negative
2
ie x = 16 means x = 4 or -4
For GMAT √x equals what?
Only the positive square root
ie √16 = 4 only
√x expressed as an exponent
1/2
x
a
√x expressed as an exponent
1/a
x
a
√xⁿ expressed as an exponent
n/a
x
√a/b expressed as a fraction
√a / √b
√a*b expressed as a fraction
√a * √b
Can you multiply or divide an inequality by a negative
Yes, but you must flip the direction of the sign
ex. 2 > 1
- 2 < -1
Express
2 < x+2 < 4
as two inequalities
It means
x+2 > 2 AND x+2 < 4
Find the range of possible values for y
y < x+2 AND y > (x-1)/2
(x-1)/2 < y < x+2
Can you add two inequalities together?
Yes - if the signs are going in the same direction
What is |x|
x or -x
ex. |x+5| =10
means x = 5 OR x = -15
Quadratics
Recognizing when you are dealing with quadratics and factoring them is a key GMAT skill
Factor:
x² = 81
(x+9)(x-9)=0
Quadratics
x² + (a+b)x + ab = 0
(x+a)(x+b) = 0
ex. x² + 2x - 8 = 0
(a+b) = 2
(a*b) = -8
(x+4)(x-2)=0
When should you factor?
● When you see addition or subtraction of roots, exponents, or factorials
● When you can remove a variable
● When you can change a mix of variables
Exponents
Which exponents have a units digit that repeats every time
0
1
5
6
Exponents
Which exponents have a units digit that repeats every 2 times
4
9
Exponents
Which exponents have a units digit that repeats every 4 times
2
3
7
8
Factoring
Remember to factor out smallest root
Factor: 2¯³ + 2¯² + 2¯⁵
2¯⁵ ( 2² + 2³ + 1)
Identify the range of values for the expression
|x| < y
-y < x < y
Identify the range of values for the expression
|x| > y
x < -y or x > y
What are the properties of 0
● It is even
● It is neither positive or negative - it is non-negative
● It is a multiple of every number
● It is a factor of no number
What numbers when squared become smaller?
All numbers between 0 & 1
