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15 Cards in this Set

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Absolute Value
Absolute value makes a negative number positive. Positive numbers and 0 are left unchanged. The absolute value of x is written |x|. We write |–6| = 6 and |8| = 8.
Absolute Value Rules
Algebra rules for absolute values are listed below.



Piecewise Definition:
Square root definition:
Rules: 1. |–a| = |a|

2. |a| ≥ 0

3. Products: |ab| = |a||b|

4. Quotients: |a / b| = |a| / |b|

5. Powers: |an| = |a|n

6. Triangle Inequality:
|a + b| ≤ |a| + |b|

7. Alternate Triangle Inequality:
|a – b| ≥ |a| – |b|

CAREFUL!! Sums:
Differences:
|a + b| is not the same as |a| + |b|
|a – b| is not the same as |a| – |b|
Algebra
The mathematics of working with variables.
Conditional Equation
An equation that is true for some value(s) of the variable(s) and not true for others.



Example: The equation 2x – 5 = 9 is conditional because it is only true for x = 7. Other values of x do not satisfy the equation.
Conditional Inequality
An inequality that is true for some value(s) of the variable(s) and not true for others.



Example: The inequality 2x – 5 < 9 is conditional because it is only true for values of x such that x < 7. Other values of x do not satisfy the inequality.
Exponent
x in the expression ax. For example, 3 is the exponent in 23.
Exponent Rules
Algebra rules and formulas for exponents are listed below.



Definitions

1. an = a·a·a···a (n times)

2. a0 = 1 (a ≠ 0)

3. (a ≠ 0)

4. (a ≥ 0, m ≥ 0, n > 0)



Combining

1. multiplication: axay = ax + y

2. division: (a ≠ 0)

3. powers: (ax)y = axy



Distributing (a ≥ 0, b ≥ 0)

1. (ab)x = axbx

2. (b ≠ 0)



Careful!!

1. (a + b)n ≠ an + bn

2. (a – b)n ≠ an – bn
Identity (Equation or Inequality)
An equation which is true regardless of what values are substituted for any variables (if there are any variables at all).



Identities: 1 + 1 = 2

(x + y)2 = x2 + 2xy + y2

a2 ≥ 0
Inequality
Definition 1: Any of the symbols <, >, ≤, or ≥.

Definition 2: A mathematical sentence built from expressions using one or more of the symbols <, >, ≤, or ≥.



Examples: x + y < 1

4 ≤ a ≤ 7

m2 – 3m + 2 ≥ 0
Transitive Property of Inequalities
If a < b and b < c , then a < c.
If a ≤ b and b ≤ c , then a ≤ c.
If a > b and b > c , then a > c.
If a ≥ b and b ≥ c , then a ≥ c.
Interval Notation
A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. For example, [3, 8) is the interval of real numbers between 3 and 8, including 3 and excluding 8.
Like Terms
Terms which have the same variables and corresponding powers and/or roots. Like terms can be combined using addition an subtraction. Terms that are not like cannot be combined using addition or subtraction.



Example: 5x2y and 8x2y are like terms.

5x2y + 8x2y simplifies to 13x2y
5x2y – 8x2y simplifies to –3x2y
Linear Equation
An equation that can be written in the form "linear polynomial = linear polynomial" or "linear polynomial = constant".

The following are examples of linear equations: 2x – 3 = 5, 4a + 9 = 8 – 9a, and 2x + 5y = 1.
Linear Inequality
An inequality that can be written in the form "linear polynomial > linear polynomial" or "linear polynomial > constant". The > sign may be replaced by <, ≤, or ≥.

The following are examples of linear inequalities: 2x – 3 < 5, 4a + 9 ≥ 8 – 9a, and 2x + 5y ≤ 1.
Scientific Notation
A standardized way of writing real numbers. In scientific notation, all real numbers are written in the form a·10b, where 1 ≤ a < 10 and b is an integer. For example, 351 is written 3.51·102 in scientific notation.