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Properties and Formulas

Chapter 1 Foundations for Algebra

Order of Operations

1. Perform an operation(s) inside grouping symbols.
2. Simplify powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Commutative Property of Addition

For every real number a and b , a + b = b + a . b comma eh plus b equals b plus eh .

Commutative Property of Multiplication

For every real number a and b , a · b = b · a . b comma eh middle dot b equals b middle dot eh .

Associative Property of Addition

For every real number a, b, and c , ( a + b ) + c = a + ( b + c ) . c comma open eh plus b close plus c equals eh plus open b plus c close .

Associative Property of Multiplication

For every real number a, b, and c , ( a · b ) · c = a · ( b · c ) . c comma open eh middle dot b close middle dot c equals eh middle dot open b middle dot c close .

Identity Property of Addition

For every real number a , a + 0 = a . eh comma eh plus 0 equals eh .

Identity Property of Multiplication

For every real number a , 1 · a = a . eh comma 1 middle dot eh equals eh .

Multiplication Property of − 1 negative 1

For every real number a , − 1 · a = − a . eh comma negative 1 middle dot eh equals negative eh .

Zero Property of Multiplication

For every real number a , a · 0 = 0 . eh comma eh middle dot 0 equals 0 .

Inverse Property of Addition

For every real number a, there is an additive inverse − a negative eh  such that a + ( − a ) = 0 . eh plus open negative eh close equals 0 .

Inverse Property of Multiplication

For every nonzero number a, there is a multiplicative inverse such that a ⋅ 1 a = 1. eh dot , 1 over eh , equals 1.

Distributive Property

For every real number a, b, and c:

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

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

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

Chapter 2 Solving Equations

Addition Property of Equality

For every real number a, b, and c, if a = b ,  then  a + c = b + c . eh equals b comma , then , eh plus c equals b plus c .

Subtraction Property of Equality

For every real number a, b, and c, if a = b ,  then  a − c = b − c . eh equals b comma , then , eh minus c equals b minus c .

Multiplication Property of Equality

For every real number a, b, and c, if a = b ,  then  a · c = b · c . eh equals b comma , then , eh middle dot c equals b middle dot c .

Division Property of Equality

For every real number a, b, and c, where c ≠ 0 , c not equal to 0 comma  if a = b , eh equals b comma  then a c = b c . eh over c , equals , b over c , .

Cross Products of a Proportion

If a b = c d , eh over b , equals , c over d , comma  then a d = b c . eh d equals b c .

Percent Proportion

a b = p 100 , eh over b , equals , p over 100 , comma  where b ≠ 0 . b not equal to 0 .

Percent Equation

a = p % · b , eh equals p percent middle dot b comma  where b ≠ 0 . b not equal to 0 .

Simple Interest Formula

I = p r t i equals p r t

Percent of Change

p % = amount of increase or decrease original amount p percent equals . fraction amountofincreaseordecrease , over originalamount end fraction

amount  of  increase  =  new  amount  −  original  amount amount . of . increase . equals , new . amount , minus . original . amount

amount  of  decrease  =  original  amount  −  new  amount amount . of . decrease . equals . original . amount , minus , new . amount

Relative Error

relative error = | measured or estimated value − actual value | actual value relativeerror . equals . fraction absolute value of . measuredorestimatedvalue . minus . actualvalue , end absolute value , , over actualvalue end fraction

Chapter 3 Solving Inequalities

The following properties of inequality are also true for ≥  and  ≤ . greater than or equal to , and , less than or equal to .

Addition Property of Inequality

For every real number a, b, and c,
if a > b , eh greater than b comma  then a + c > b + c ; eh plus c greater than b plus c semicolon
if  a < b , if eh less than b comma  then a + c < b + c . eh plus c less than b plus c .

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