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Properties and Formulas

Order of Operations

1. Perform any operation(s) inside grouping symbols.
2. Simplify any terms with exponents.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.

The Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a 2 + b 2 = c 2 eh squared , plus , b squared , equals , c squared

The Distance Formula

The distance d between any two points ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close and ( x 2 , y 2 ) open , x sub 2 , comma , y sub 2 , close is d = d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . d equals . d equals . square root of open . x sub 2 , minus , x sub 1 . close squared . plus . open . y sub 2 , minus , y sub 1 . close squared end root . .

The Midpoint Formula

The midpoint M of a line segment with endpoints A ( x 1 , y 1 ) eh open , x sub 1 , comma , y sub 1 , close and B ( x 2 , y 2 ) b open , x sub 2 , comma , y sub 2 , close is ( x 1 + x 2 2 , y 1 + y 2 2 ) . open . fraction x sub 1 , plus , x sub 2 , over 2 end fraction . comma . fraction y sub 1 , plus , y sub 2 , over 2 end fraction . close . .

Chapter 1 Expressions, Equations, and Inequalities

Closure

For all real numbers a and b, a + b and a · b eh middle dot b are real numbers.

The Associative Properties

For all real numbers a, b, and c:

( a + b ) + c = a + ( b + c ) open eh plus b close plus c equals eh plus open b plus c close

( a · b ) · c = a · ( b · c ) open eh middle dot b close middle dot c equals eh middle dot open b middle dot c close

The Commutative Properties

For all real numbers a and b:

a + b = b + a  and  a · b = b · a eh plus b equals b plus eh , and , eh middle dot b equals b middle dot eh

The Identity Properties

For every real number a:

a + 0 = 0 equals a and 0 + a = a 0 plus eh equals eh	a · 1 = a  and  1 · a = a eh middle dot , 1 equals eh , and , 1 middle dot eh equals eh
0 is the additive identity.	1 is the multiplicative identity.

The Inverse Properties

For every real number a:

a + ( − a ) = 0  and  a ⋅ 1 a = 1 ( a ≠ 0 ) eh plus open negative eh close equals 0 , and . eh dot , 1 over eh , equals 1 . open eh not equal to 0 close

The Distributive Properties

For all real numbers a, b, and c:

a(b + c ) = close equals ab + ac	(b + c) a = eh equals ba + ca
a ( b − c ) = a b − a c eh open b minus c close equals eh b minus eh c	( b − c ) a = b a − c a open b minus c close eh equals b eh minus c eh

Multiplication

Let a represent a real number.

Multiplication by 0 : 0 · a = 0 . 0 colon 0 middle dot eh equals 0 .

Multiplication by − 1 : − 1 · a = − a negative 1 colon negative 1 middle dot eh equals negative eh

Opposites

Let a and b represent real numbers.

Opposite of a Sum:	− ( a + b ) = − a + ( − b ) = − a − b negative open eh plus b close equals negative eh plus open negative b close equals negative eh minus b
Opposite of a Difference:	− ( a − b ) = − a + b = b − a negative open eh minus b close equals negative eh plus b equals b minus eh
Opposite of a Product:	− ( a b ) = − a · b = a · ( − b ) negative open eh b close equals negative eh middle dot b equals eh middle dot open negative b close
Opposite of an Opposite:	− ( − a ) = a negative open negative eh close equals eh

Properties of Equality

Assume a, b, and c represent real numbers.

Reflexive:	a = a eh equals eh
Symmetric:	If a = b , eh equals b comma then b = a . b equals eh .
Transitive:	If a = b eh equals b and b = c , b equals c comma then a = c . eh equals c .
Substitution:	If a = b , eh equals b comma then you can replace a with b and vice versa.
Addition:	If a = b , eh equals b comma then a + c = b + c . eh plus c equals b plus c .
Subtraction:	If a = b , eh equals b comma then a − c = b − c . eh minus c equals b minus c .
Multiplication:	If a = b , eh equals b comma then a c = b c . eh c equals b c .
Division:	If a = b  and  c ≠ 0 , eh equals b , and , c not equal to 0 comma then a c = b c . eh over c , equals , b over c , .

Properties of Inequality

Let a, b, and c represent real numbers.

Transitive:	If a > b eh greater than b and b > c , b greater than c comma then a > c . eh greater than c .
Addition:	If a > b , eh greater than b comma then a + c > b + c . eh plus c greater than b plus c .
Subtraction:	If a > b , eh greater than b comma then a − c > b − c . eh minus c greater than b minus c .
Multiplication:	If a > b eh greater than b and c > 0 , c greater than 0 comma then a c > b c . eh c greater than b c . If a > b eh greater than b and c < 0 , c less than 0 comma then a c < b c . eh c less than b c .
Division:	If a > b eh greater than b and c > 0 , c greater than 0 comma then a c > b c . eh over c , greater than , b over c , . If a > b eh greater than b and c < 0 , c less than 0 comma then a c < b c . eh over c , less than , b over c , .

Chapter 2 Functions, Equations, and Graphs

Direct Variation

y = k x y equals k x or y x = k , y over x , equals k comma where k ≠ 0 . k not equal to 0 .

Slope of a Line Containing ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close and ( x 2 , y 2 ) x sub 2 , comma , y sub 2 , close

slope = vertical change ( rise ) horizontal change ( run ) = y 2 − y 1 x 2 − x 1 , slope , equals . fraction verticalchange . open , rise , close , over horizontalchange . open run close end fraction . equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction . comma

where x 2 − x 1 ≠ 0 x sub 2 , minus , x sub 1 , not equal to 0

Point-Slope Equation of a Line

The equation of the line through point ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close with slope m is y − y 1 = m ( x − x 1 ) . y minus , y sub 1 , equals m open x minus , x sub 1 , close .

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