## Key Concepts

- Phrases such as
*sum*,*increased by*,*difference*,*decreased by*,*of*,*an*d, etc. can be translated into mathematical operations and notation to help solve a problem. - To add or subtract fractions, make sure they each have the same denominator first.
- When dividing fractions, use the phrase
*keep-change-flip*to remind you to multiply the first fraction by the reciprocal of the second one. - Just like the digits of a whole number increase to the left by powers of 10, the digits after the decimal on the right decrease by fractions of powers of ten.
- When using inequality symbols, the smaller side of the symbol faces the smaller number and the larger side faces the larger number.
- The product or quotient of two negative numbers or two positive numbers is always positive; the product or quotient of two differently signed numbers is always negative.

## Key Expressions, Equations, and Inequalities

- [latex]a=b[/latex] is read as “[latex]a[/latex] is equal to [latex]b[/latex].”
- [latex]a\neq b[/latex] is read as “[latex]a[/latex] is not equal to [latex]b[/latex].”
- [latex]a\gt b[/latex] can be read “[latex]a[/latex] is greater than [latex]b[/latex]” or “[latex]b[/latex] is less than [latex]a[/latex].”
- A fraction is written [latex]\dfrac{a}{b},[/latex] where [latex]a[/latex] and [latex]b[/latex] are integers and [latex]b \neq 0[/latex]. In a fraction, [latex]a[/latex] is called the numerator and [latex]b[/latex] is called the denominator.
- The distributive property, [latex]a(b+c)=ab+ac[/latex], represents the distribution of multiplication over addition or subtraction.
- The property of one states that any number, except zero, divided by itself is one. That is [latex]\dfrac{a}{a}=1[/latex], where [latex]a \neq 0[/latex].
- if [latex]a, \text{ } b, \text{ and } c[/latex] are numbers such that [latex]b \neq 0, \text { } c \neq 0,[/latex] then [latex]\dfrac{a}{b} = \dfrac{a\cdot c}{b\cdot c}[/latex].
- To multiply fractions, [latex] \frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}=\frac{\text{product of the numerators}}{\text{product of the denominators}}[/latex]

## Glossary

**absolute value**- a number’s distance from zero on the number line, which is always positive
**constant**- a number whose value always stays the same
**distributive property**- a number multiplying an expression inside of parentheses distributes to each term in the contained expression
**equation**- two expressions connected by an equal sign
**equivalent fractions**- fractions that have the same value
**exponent**- a number in a superscript position that tells how many times to multiply the base by itself
**expression**- a number, a variable, or a combination of numbers and variables and operation symbols
**improper fraction**- a fraction [latex]\dfrac{a}{b}, \text{ } b \neq 0[/latex] is proper if [latex]a \lt b[/latex], and is improper if [latex]a \geq b.[/latex]
**inequality**- two expressions connected by an inequality sign
**integers**- counting numbers like 1, 2, 3, … including their opposites (negatives) and zero
**like terms**- terms where the variables match exactly (exponents included)
**operators**- symbols that represent arithmetic operations such as addition, subtraction, multiplication, and division

**order of operations**- the universally accepted order to perform operations when more than one is present in an expression, often represented by the acronym PEMDAS
**real numbers**- fractions, negative numbers, decimals, integers, square roots, and zero
**reciprocal**- two fractions are reciprocals if their product is [latex]1[/latex]
**simplified fraction**- also called a reduced fraction, or a fraction in lowest terms, a fraction having no common factors in the numerator and denominator
**term**- a single number, variable, or a product or quotient of numbers and/or variables
**variable**- a letter that represents a number or quantity whose value may change