Summary: Review

Key Concepts

	Addition	Multiplication
Commutative Property	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative Property	$a+\left(b+c\right)=\left(a+b\right)+c$	$a\left(bc\right)=\left(ab\right)c$
Distributive Property	$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$
Identity Property	There exists a unique real number called the additive identity, 0, such that, for any real number a $a+0=a$	There exists a unique real number called the multiplicative identity, 1, such that, for any real number a $a\cdot 1=a$
Inverse Property	Every real number a has an additive inverse, or opposite, denoted $–a$ , such that $a+\left(-a\right)=0$	Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted $\Large\frac{1}{a}$ , such that $a\cdot \left(\Large\frac{1}{a}\normalsize\right)=1$

We simplify an expression by removing grouping symbols and combining like terms.
Properties of Equality For two expressions S and T and any constant c,
- Addition Property of Equality: If $S=T$ then $S+c=T+c$
- Multiplication Property of Equality: If $S=T$ then $S \cdot c = T \cdot c$ , provided $c \neq 0$ .
To solve a multi-step equation
- Multiply to clear any fractions or decimals (optional)
- Simplify each side by clearing parentheses and combining like terms.
- Add or subtract to isolate the variable term—possibly a term with the variable.
- Multiply or divide to isolate the variable.
The the solutions of $|x|=a$ is $x = a$ or $x = -a$

For any positive value of a and x, a single variable, or any algebraic expression:

Absolute Value Inequality	Equivalent Inequality	Interval Notation
$\left\|{ x }\right\|\le{ a}$	${ -a}\le{x}\le{ a}$	$\left[-a, a\right]$
$\left\| x \right\|\lt{a}$	${ -a}\lt{x}\lt{ a}$	$\left(-a, a\right)$
$\left\| x \right\|\ge{ a}$	${x}\le\text{−a}$ or ${x}\ge{ a}$	$\left(-\infty,-a\right]\cup\left[a,\infty\right)$
$\left\| x \right\|\gt\text{a}$	$\displaystyle{x}\lt\text{−a}$ or ${x}\gt{ a}$	$\left(-\infty,-a\right)\cup\left(a,\infty\right)$

coefficient: constant factor in a term
like terms: have exactly the same variable factors
absolute value: the absolute value of a number $n$ , written $|n|$ , is its distance from $0$ on the number line. $|n| \geq 0$ for every real number $n$ .