# ACT Pre-Algebra Cheat Sheet

Need a quick refresh on the basics of Pre-Algebra tested on the ACT? Print out this handy cheat sheet and review it often to quickly improve your ACT Math foundation. The more you practice, the better your scores will be!

### Basic Definitions

A **variable** is a symbol representing a numerical quantity. Variables are represented by letters in the alphabet such as x, a, b, y, etc. The number that the variable represents is called a **value**.

A **constant** is a symbol that represents a definite quantity (such as pi).

A **term** is a product (multiplication) with an unspecified number of factors, made up of either variables or constants. Terms that have the same factors which differ only in their numerical coefficients are called **similar terms**. For example, 5y and 9y are similar terms.

An **algebraic expression** is a mathematical statement which often uses constants and variables. For example: 75*x* + 12

Because of the distributive property, similar terms can be combined into one term. The new term has the same factors as the similar terms, but its coefficient is the sum (addition) of the coefficients; this is commonly known as combining like terms. 3*xy* + 2*xy* = 5*xy*

### Concepts

**Subtracting negative numbers**: When you subtract a negative number, you will add the terms. Example: 5 – (-2) = 5 + 2 = 7

**Zero**: Dividing by zero is undefined. The denominator of a fraction cannot be zero. 1/0 = undefined.

**Recpirocals**: The reciprocal of a number is 1 divided by the number. For a fraction, the reciprocal can be determined by flipping the numerator and the denominator. A number times its reciprocal = 1.

**Squaring Fractions**: If you square a number between 0 and 1, the number gets *smaller*. For example, (1/2)^{2} =1/4.

**Multiplying & dividing negatives**: The product or quotient of two numbers with the same sign is positive, even if both numbers are negative. (-4) x (-2) = 8.

**Fundamental Counting Principle**: if an event has *x* possible outcomes and another independent event has *y* possible outcomes, then there are *xy* possible ways the two events could occur together.

**Probability**: The probability = # of desired outcomes / # of total possible outcomes

### Statistics Basics

**Domain:** The inputs of a function; all possible values for x.

**Range**: The outputs of a function; all possible values for f(x). For a set of data, the range is found by subtracting the smallest number from the largest number.

**Mean:** The mathematical average. This is defined as the sum of the terms divided by the number of terms. For a list of consecutive integers or evenly spaced numbers, the mean and the median are equal.

**Median**: The middle number when the set is ordered numerically. For a set with an even number of terms, the median is the average of the two middle numbers.

**Mode**: The number that occurs most frequently in a set. A set can have no mode ({2,3,4,5} for example), or it can have more than one mode ({2,2,3,3,4,5} has 2 and 3 as he modes}).

**Inqualities: **The basic symbols for inequalities are:

< less than

Example: x < 7 means all numbers less than 7.

> greater than

Example: x > 4 means that x can be all numbers greater than 4.

≤ less than or equal to

Example: x ≤ ½ means that x can be ½ or any number less than ½

≥ greater than or equal to

Example: x ≥ 0 means that x can equal 0 or be any number greater than 0

**Absolute value:** This will always the a positive number, because the absolute value represents the distance from 0 on a number line, and a distance cannot be negative. For x > 0, |x| = x, and for x < 0, |x| = -x.

**Solving inequalities + absolute values**: To solve absolute value equations, we must split them into two separate equations, removing the absolute value, and making one equation negative. There will always be TWO solutions. Let’s look at an example.