Pre-Algebra Master Workbook | 20 Core Problems

Unit 01

Integers & Absolute Value

Integers include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …

Rules for operations:

(+) × (+) = (+) (−) × (−) = (+)
(+) × (−) = (−) (−) × (+) = (−)
|a| = a if a ≥ 0 ; |a| = −a if a < 0

Key terms: absolute value opposite integers number line

Worked Example

Evaluate: −8 + (−3) × 2 − |−5|

Step 1: (−3) × 2 = −6
Step 2: |−5| = 5
Step 3: −8 + (−6) − 5 = −19
∴ Answer = −19

Unit 02

Fractions, Decimals & Mixed Numbers

A fraction a/b means a ÷ b. To add/subtract fractions, find a common denominator.

a/b + c/d = (ad + bc) / bd
a/b × c/d = (a×c) / (b×d)
a/b ÷ c/d = a/b × d/c (multiply by reciprocal)

LCD improper fraction reciprocal

Worked Example

Simplify: 2/3 + 5/6 − 1/4

LCD = 12
8/12 + 10/12 − 3/12 = 15/12 = 5/4 = 1 1/4

Unit 03

Ratios & Proportions

A ratio compares two quantities. A proportion states two ratios are equal.

a/b = c/d ⟺ ad = bc (Cross-multiplication)
Unit rate: divide to find quantity per 1 unit

cross multiply unit rate scale factor

Worked Example

Solve: 3/5 = x/20

Cross multiply: 5x = 60
x = 60 ÷ 5 = 12

Unit 04

Percents

Percent means "per hundred." Conversion triangle:

Part = Percent × Whole (P = r × W)
% change = (new − old) / old × 100
Discount = original × rate
Tax / Tip = base × percent

percent increase percent decrease markup

Worked Example

A shirt costs $40. It's on sale 25% off. What is the sale price?

Discount = 40 × 0.25 = $10
Sale price = 40 − 10 = $30

Unit 05

Algebraic Expressions

An expression contains variables, numbers, and operations — but no equal sign.

Distributive: a(b + c) = ab + ac
Combine like terms: 3x + 5x = 8x
Order of Operations: PEMDAS
Parentheses → Exponents → ×÷ → +−

variable coefficient constant like terms

Worked Example

Simplify: 3(2x − 4) + 5x

= 6x − 12 + 5x
= 11x − 12

Unit 06

Solving One & Two-Step Equations

Isolate the variable by applying inverse operations to both sides equally.

One-step: x + a = b → x = b − a
Two-step: ax + b = c → x = (c − b) / a
Check: always substitute back to verify!

inverse operation balance method solution

Worked Example

Solve: 4x − 7 = 13

+7 both sides: 4x = 20
÷4 both sides: x = 5

Unit 07

Inequalities

Solve like equations, but flip the inequality sign when multiplying or dividing by a negative number.

Symbols: < > ≤ ≥ ≠
If −2x > 6, then x < −3 (sign flips!)
Graph: open circle for < or >, closed for ≤ or ≥

flip sign rule compound inequality solution set

Worked Example

Solve and graph: −3x + 2 ≤ −7

−3x ≤ −9
÷ (−3), flip sign: x ≥ 3
∴ x ≥ 3

Unit 08

Exponents & Scientific Notation

a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m−n)
(a^m)^n = a^(m×n)
a^0 = 1 (a ≠ 0)
Scientific: a × 10^n where 1 ≤ a < 10

base exponent power rules scientific notation

Worked Example

Simplify: (2x²)³ × x⁻¹

= 8x⁶ × x⁻¹
= 8x⁵

Unit 09

Coordinate Plane & Graphing

The coordinate plane has x-axis (horizontal) and y-axis (vertical), divided into 4 quadrants.

Distance = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint = ((x₁+x₂)/2 , (y₁+y₂)/2)
Quadrant I: (+,+) II: (−,+)
Quadrant III: (−,−) IV: (+,−)

ordered pair quadrant origin

Worked Example

Find the midpoint of A(2, −4) and B(8, 6).

M = ((2+8)/2 , (−4+6)/2)
= (5, 1)

Unit 10

Functions & Linear Relationships

A function assigns exactly one output to each input. A linear function has a constant rate of change (slope).

Slope m = (y₂ − y₁) / (x₂ − x₁)
Slope-intercept: y = mx + b
b = y-intercept (where line crosses y-axis)

slope y-intercept input/output rate of change

Worked Example

Find the slope between (1, 3) and (4, 12).

m = (12−3)/(4−1) = 9/3 = 3

Unit 11

Statistics & Data Analysis

Mean = sum of data / number of values
Median = middle value (sorted)
Mode = most frequent value
Range = max − min
MAD = mean of |x − mean|

mean median mode range outlier

Worked Example

Find the mean: {4, 7, 9, 13, 7}

Sum = 4+7+9+13+7 = 40
Mean = 40 ÷ 5 = 8

Unit 12

Geometry: Area, Volume & Angles

Rectangle: A = l × w P = 2(l+w)
Triangle: A = (1/2) × b × h
Circle: A = πr² C = 2πr
Rectangular prism: V = l × w × h
Angles: Supplementary = 180° ; Complementary = 90°

perimeter area volume π ≈ 3.14

Worked Example

Find the area of a triangle with base 10 cm and height 6 cm.

A = (1/2) × 10 × 6 = 30 cm²