Pre-Algebra – Complete Reference Guide

Integers & The Number Line

Operations with Integers

Adding Same Signs: Add absolute values, keep sign.

$3 + 5 = 8$

$(-3) + (-5) = -8$

Adding Different Signs: Subtract smaller $|val|$ from larger, keep sign of larger.

$-8 + 5 = -3$

$8 + (-5) = 3$

Subtraction: "Keep Change Change" (Add the opposite).

$5 - (-3) \Rightarrow 5 + 3 = 8$

Multiplying & Dividing

Same Signs $\Rightarrow$ Positive

$(+) \cdot (+) = +$
$(-) \cdot (-) = +$

Different Signs $\Rightarrow$ Negative

$(+) \cdot (-) = -$
$(-) \cdot (+) = -$

Absolute Value

Distance from zero (always positive).

$|5| = 5$

$|-5| = 5$

Algebraic Expressions

Vocabulary

Variable: Symbol for unknown ($x, y, z$).

Coefficient: Number multiplying variable ($5x \to 5$).

Constant: Number without variable ($5x + 3 \to 3$).

Term: Parts separated by $+$ or $-$ signs.

Simplifying

Combine Like Terms: Add/sub coefs of terms with same variable AND exponent.

$3x + 2y - x + 5y = 2x + 7y$

Distributive Property

$a(b+c) = ab + ac$

$3(x+4) = 3x + 12$

$-(x-5) = -x + 5$

Solving Equations

Goal

Isolate the variable. Do the inverse operation to BOTH sides.

One-Step Equations

$x + 5 = 12 \Rightarrow$ Subtract 5 $\Rightarrow x = 7$

$3x = 15 \Rightarrow$ Divide by 3 $\Rightarrow x = 5$

Two-Step Equations

1. Undo Addition/Subtraction first.

2. Undo Mult/Division second.

$2x + 3 = 13$

Subtract 3: $2x = 10$

Divide by 2: $x = 5$

Inequalities

Symbols

$<$ Less than

$>$ Greater than

$\leq$ Less than or equal to

$\geq$ Greater than or equal to

The "Golden Rule"

When you MULTIPLY or DIVIDE by a NEGATIVE number, you must FLIP the inequality sign!

Ex: $-2x > 10 \Rightarrow x < -5$

Graphing on Number Line

Open circle $\circ$ for $<,>$. Closed circle $\bullet$ for $\leq, \geq$.

Exponents & Roots

$base^{exponent}$: $5^3 = 5 \cdot 5 \cdot 5 = 125$

Key Rules

$x^a \cdot x^b = x^{a+b}$

$(x^a)^b = x^{a \cdot b}$

$x^0 = 1$ (if $x \neq 0$)

Scientific Notation

$a \times 10^n$ ($1 \leq a < 10$)

$3,500 = 3.5 \times 10^3$

$0.0042 = 4.2 \times 10^{-3}$

Square Roots

$\sqrt{25} = 5$ because $5^2 = 25$

Perfect squares: 1, 4, 9, 16, 25, 36...

Linear Equations Intro

Slope-Intercept Form

$y = mx + b$

$m$ = Slope (Rate of Change)

$b$ = y-intercept (Start Value)

Calculating Slope

$m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}$

Basic Geometry

Angles

Acute: Less than $90^\circ$

Right: $=90^\circ$ (Little box)

Obtuse: Between $90^\circ$ and $180^\circ$

Straight: $=180^\circ$

Complementary: Sum to $90^\circ$

Supplementary: Sum to $180^\circ$

Triangles (Sum $180^\circ$)

Scalene: 0 equal sides

Isosceles: 2 equal sides

Equilateral: 3 equal sides

Statistics & Probability Intro

Measures of Center

Mean (Average): Sum of values ÷ Count

Median: Middle value (when ordered)

Mode: Most frequent value

Range: Max - Min

Basic Probability

$P(\text{Event}) = \frac{\text{# Favorable Outcomes}}{\text{Total Possible Outcomes}}$

Example: Rolling a 3 on a die

$P(3) = \frac{1}{6}$

PRE-ALGEBRA Complete Reference