Summary: Review

Key Concepts

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

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 [latex] S=T[/latex] then [latex]S+c=T+c[/latex]
- Multiplication Property of Equality: If [latex]S=T[/latex] then [latex]S \cdot c = T \cdot c[/latex], provided [latex]c \neq 0[/latex].
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 [latex]|x|=a[/latex] is [latex]x = a[/latex] or [latex]x = -a[/latex]

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

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

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