Master Series · Vol. 1

Algebra 1
Workbook

20 Core Problems · All Major Units · Instant Feedback

Real Numbers Expressions Equations Inequalities Functions Linear Functions Systems Exponents Polynomials Factoring Quadratics Statistics

⏱ 40 MIN  |  20 PROBLEMS  |  SHORT ANSWER

Concept Review & Key Formulas

Review each unit before attempting the problems

1

Real Numbers & Number Sets

Real numbers include naturals (ℕ), whole numbers, integers (ℤ), rationals (ℚ), and irrationals. Every real number is either rational or irrational.

★ Memorize
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
Rational: p/q form (q ≠ 0)
Irrational: non-terminating, non-repeating decimal
|a| = a if a ≥ 0; |a| = −a if a < 0
Example
Classify: −4, 0.75, √5, 0, −3/7
−4 → Integer; 0.75 → Rational; √5 → Irrational; 0 → Whole; −3/7 → Rational
2

Algebraic Expressions & Operations

An algebraic expression contains variables, coefficients, and constants combined with operations. Like terms share identical variable parts and can be combined.

★ Memorize
Distributive: a(b + c) = ab + ac
Combine like terms: 3x + 5x = 8x
Evaluate: substitute value for variable
Example
Simplify: 4x − 3 + 2x + 7
= (4x + 2x) + (−3 + 7) = 6x + 4
3

Linear Equations (One Variable)

A linear equation in one variable has degree 1. Solve by isolating the variable using inverse operations (addition↔subtraction, multiplication↔division).

★ Memorize
Standard form: ax + b = c
Steps: Distribute → Combine → Move vars → Isolate
Check: substitute answer back
Example
Solve: 3x − 5 = 16
3x = 21 → x = 7
4

Linear Inequalities

Inequalities use <, >, ≤, ≥. Solve like equations, but FLIP the inequality sign when multiplying or dividing by a NEGATIVE number.

★ KEY RULE — Flip when × or ÷ negative
−2x > 8 → x < −4 (sign flipped!)
Compound: a < x < b (interval notation)
Graph: open circle (</>), closed circle (≤/≥)
Example
Solve: −3x + 4 ≤ 13
−3x ≤ 9 → x ≥ −3
5

Functions & Graphs

A function maps each input (domain) to exactly ONE output (range). Tested with the Vertical Line Test on a graph.

★ Memorize
f(x) = expression in x
Domain: set of valid inputs
Range: set of all outputs
Vertical Line Test: a graph is a function if no vertical line crosses it more than once
Example
f(x) = 2x − 3. Find f(5).
f(5) = 2(5) − 3 = 10 − 3 = 7
6

Linear Functions & Slope

A linear function forms a straight line. Slope (m) measures steepness. The y-intercept (b) is where the line crosses the y-axis.

★ Memorize All Three Forms
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Slope-Intercept: y = mx + b
Standard Form: Ax + By = C
Point-Slope: y − y₁ = m(x − x₁)
Parallel: equal slopes · Perpendicular: m₁ × m₂ = −1
Example
Find slope between (2, 5) and (6, 13)
m = (13−5)/(6−2) = 8/4 = 2
7

Systems of Linear Equations

A system is two or more equations with the same variables. The solution is the point of intersection — values that satisfy ALL equations simultaneously.

★ Three Methods
Substitution: solve one eq for a var, substitute
Elimination: add/subtract equations to cancel a var
Graphing: find intersection point
No solution: parallel lines · Infinite: same line
Example
Solve: x + y = 7 and x − y = 3
Add: 2x = 10 → x = 5, y = 2. Solution: (5, 2)
8

Laws of Exponents

Exponent rules govern how to simplify expressions with powers. These rules apply to all real bases (b ≠ 0).

★ Memorize All Rules
Product: bᵐ × bⁿ = bᵐ⁺ⁿ
Quotient: bᵐ ÷ bⁿ = bᵐ⁻ⁿ
Power: (bᵐ)ⁿ = bᵐⁿ
Zero exp: b⁰ = 1
Negative exp: b⁻ⁿ = 1/bⁿ
Example
Simplify: x³ × x⁵ ÷ x²
= x^(3+5−2) = x⁶
9

Polynomials: Addition, Subtraction & Multiplication

A polynomial is a sum of terms with non-negative integer exponents. To add/subtract, combine like terms. To multiply, use the Distributive Property (FOIL for binomials).

★ FOIL Method
FOIL: (a+b)(c+d) = ac + ad + bc + bd
(a+b)² = a² + 2ab + b²
(a+b)(a−b) = a² − b²
Degree = highest exponent
Example
Expand: (x + 3)(x − 5)
= x² − 5x + 3x − 15 = x² − 2x − 15
10

Factoring Polynomials

Factoring reverses multiplication. Always look for GCF first, then try special patterns or the ac method.

★ Factoring Priority Order
1. GCF first
2. Difference of Squares: a²−b² = (a+b)(a−b)
3. Trinomial: x²+bx+c → find two numbers that multiply to c and add to b
4. Perfect Square: a²+2ab+b² = (a+b)²
Example
Factor: x² + 5x + 6
Find two numbers × to 6, + to 5: (2)(3)=6, 2+3=5
Answer: (x + 2)(x + 3)
11

Quadratic Equations

A quadratic equation has degree 2. It can have 0, 1, or 2 real solutions (roots). Use the Quadratic Formula when factoring is not possible.

★ Quadratic Formula — Must Memorize
Standard form: ax² + bx + c = 0
x = (−b ± √(b²−4ac)) / 2a
Discriminant: D = b²−4ac
D > 0: two real roots · D = 0: one root · D < 0: no real roots
Vertex: x = −b/(2a)
Example
Solve: x² − 5x + 6 = 0
Factor: (x−2)(x−3) = 0 → x = 2 or x = 3
12

Statistics: Mean, Median, Mode, Range

Descriptive statistics summarize a data set. Each measure of central tendency describes the "center" differently.

★ Memorize Definitions
Mean = sum of all values / number of values
Median = middle value when sorted
Mode = most frequently occurring value
Range = max − min
Example
Data: {4, 7, 2, 7, 10}. Find mean, median, mode, range.
Mean = 30/5 = 6 · Median = 7 · Mode = 7 · Range = 10−2 = 8
Algebra 1 · 20 Problems
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