# Algebraic Expressions, Equations, and Inequalities

## Vocabulary

number zero, which has the property that $a+0=a$ for any real number $a$

for a real number $a$, the number $-a$. The sum of a number and its additive inverse is the additive identity, zero.

property stating that when adding three or more numbers, the sum does not change based on the way the numbers are grouped:
$a+(b+c)=(a+b)+c$

### associative property of multiplication

property stating that when multiplying three or more real numbers, the product does not change based on the way the numbers are grouped:
$a(bc)=(ab)c$

property stating that when adding real numbers, the sum does not change based on the order of the numbers:
$a+b=b+a$

### commutative property of multiplication

property stating that when multiplying real numbers, the product does not change based on the order of the numbers:
$ab=ba$

### distributive property

property stating that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products:
$a(b+c)=ab+ac$

Property stating that the sum of zero and any number is the number itself:
$a+0=a$

### identity property of multiplication

property stating that the product of $1$ and any number is the given number:
$a\cdot1=a$

### multiplicative identity

number 1, which has the property $a\cdot1=a$ for any real number $a$

### multiplicative inverse

for a nonzero real number $a$, the number $\frac{1}{a}$. The product of a number and its multiplicative inverse is the multiplicative identity, 1.

### order of operations

set of rules indicating which calculations to perform first to simplify a mathematical expression