Key Concepts
- Properties of real numbers: For any real numbers a, b, and c,
| 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]
Glossary
- 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].
