🦕 Algebra 1 with Dino!

Unit 1

🔢 Real Numbers & Properties

The foundation of all algebra — understanding numbers and how they behave.

🧠 Key Concepts

Real numbers include Natural numbers (1,2,3…), Whole numbers (0,1,2…), Integers (…-2,-1,0,1,2…), Rational numbers (fractions), and Irrational numbers (√2, π). The absolute value |x| is the distance from 0 — always non-negative.

⭐ MEMORIZE THESE PROPERTIES

Commutative: a + b = b + a | a × b = b × a
Associative: (a+b)+c = a+(b+c)
Distributive: a(b+c) = ab + ac
Identity: a + 0 = a | a × 1 = a
Inverse: a + (−a) = 0 | a × (1/a) = 1

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"Dino tip: |−7| = 7 because distance is NEVER negative! Distance doesn't care about direction!"

✏️ Example

Simplify: |−12| + |5| − |−3|

|−12| = 12 (absolute value removes sign)

|5| = 5 (already positive)

|−3| = 3 (absolute value removes sign)

12 + 5 − 3 = 14

✅ Answer: 14

Unit 2

🧮 Expressions & Order of Operations

Evaluating algebraic expressions correctly using PEMDAS.

⭐ PEMDAS (Order of Operations)

Parentheses first
Exponents second
Multiplication & Division (left to right)
Addition & Subtraction (left to right)

Evaluate: 3x² + 2x − 1 when x = 4

✏️ Example

Evaluate: 3x² + 2x − 1 when x = 4

Step 1: Substitute x = 4 → 3(4²) + 2(4) − 1

Step 2: Exponent first → 3(16) + 2(4) − 1

Step 3: Multiply → 48 + 8 − 1

Step 4: Add/Subtract left to right → 55

✅ Answer: 55

Unit 3

⚖️ Solving Linear Equations

The heart of Algebra 1 — isolate the variable to find its value.

🧠 Golden Rule

Whatever you do to one side, you must do to the other side. Keep the equation balanced like a scale!

⭐ Steps to Solve

Distribute (remove parentheses)
Combine like terms on each side
Move variable terms to one side
Move constants to the other side
Divide both sides by the coefficient

✏️ Example

Solve: 3(2x − 4) = 2x + 8

Distribute: 6x − 12 = 2x + 8

Subtract 2x: 4x − 12 = 8

Add 12: 4x = 20

Divide by 4: x = 5

Check: 3(2·5−4) = 3(6) = 18 ✓ and 2(5)+8 = 18 ✓

✅ Answer: x = 5

Unit 4

📏 Linear Inequalities

Solving and graphing inequalities on a number line.

⭐ Critical Rule — FLIP THE SIGN!

When you multiply or divide both sides by a NEGATIVE number, the inequality sign FLIPS!
Example: −2x < 6 → x > −3 (sign flipped!)
Open circle ○ for < or > (not included)
Closed circle ● for ≤ or ≥ (included)

✏️ Example

Solve: −3x + 7 ≥ −8

Subtract 7 from both sides: −3x ≥ −15

Divide by −3 (FLIP the sign!): x ≤ 5

Graph: closed circle at 5, arrow pointing LEFT (←●)

✅ Answer: x ≤ 5

Unit 5

🔄 Functions & Relations

Understanding input-output relationships and function notation.

🧠 Function Definition

A function is a rule where each input (x) has exactly ONE output (y). Use the Vertical Line Test: if any vertical line crosses the graph more than once, it is NOT a function.

f(x) = 2x + 3 → f(4) = 2(4) + 3 = 11

✏️ Example

If f(x) = x² − 3x + 2, find f(−1)

Substitute x = −1: f(−1) = (−1)² − 3(−1) + 2

= 1 + 3 + 2

= 6

✅ Answer: f(−1) = 6

Unit 6

📈 Linear Functions & Slope

Graphing lines and understanding slope and intercepts.

⭐ Essential Formulas

Slope: m = (y₂ − y₁) / (x₂ − x₁) = rise/run
Slope-Intercept: y = mx + b (m=slope, b=y-intercept)
Standard Form: Ax + By = C
Point-Slope: y − y₁ = m(x − x₁)
Parallel lines: same slope (m₁ = m₂)
Perpendicular lines: slopes are negative reciprocals (m₁ × m₂ = −1)

✏️ Example

Find the slope of the line through (2, 5) and (6, 13)

m = (13 − 5) / (6 − 2)

m = 8 / 4

m = 2

✅ Answer: slope = 2

Unit 7

🔀 Systems of Equations

Finding where two lines intersect using three methods.

⭐ Three Methods

Graphing: find intersection point on graph
Substitution: solve one equation for a variable, substitute into the other
Elimination: add or subtract equations to cancel a variable
No solution → parallel lines (same slope, different y-intercept)
Infinite solutions → same line (identical equations)

✏️ Example (Substitution)

Solve: y = 2x + 1 and 3x + y = 16

Substitute y = 2x+1 into 3x + y = 16:

3x + (2x + 1) = 16 → 5x + 1 = 16 → 5x = 15 → x = 3

Back-substitute: y = 2(3) + 1 = 7

Check: 3(3) + 7 = 9 + 7 = 16 ✓

✅ Answer: (3, 7)

Unit 8

💥 Exponents & Polynomials

Mastering exponent rules and polynomial operations.

⭐ Exponent Laws

Product: xᵃ · xᵇ = x^(a+b)
Quotient: xᵃ / xᵇ = x^(a−b)
Power: (xᵃ)ᵇ = x^(ab)
Zero: x⁰ = 1 (x ≠ 0)
Negative: x^(−n) = 1/xⁿ
FOIL: (a+b)(c+d) = ac + ad + bc + bd

✏️ Example (FOIL)

Expand: (2x + 3)(x − 4)

First: 2x · x = 2x²

Outer: 2x · (−4) = −8x

Inner: 3 · x = 3x

Last: 3 · (−4) = −12

Combine: 2x² − 8x + 3x − 12 = 2x² − 5x − 12

✅ Answer: 2x² − 5x − 12

Unit 9

🧩 Factoring Polynomials

Breaking expressions into products — the reverse of expanding.

⭐ Factoring Methods

GCF first: always pull out the greatest common factor first!
Trinomial: x² + bx + c = (x + p)(x + q) where p·q = c and p+q = b
Difference of Squares: a² − b² = (a+b)(a−b)
Perfect Square: a² + 2ab + b² = (a+b)²

✏️ Example

Factor: x² + 7x + 12

Need two numbers that multiply to 12 AND add to 7

Try: 3 × 4 = 12 ✓ and 3 + 4 = 7 ✓

Therefore: (x + 3)(x + 4)

Check by FOIL: x² + 4x + 3x + 12 = x² + 7x + 12 ✓

✅ Answer: (x + 3)(x + 4)

Unit 10

🚀 Quadratic Equations & Radicals

Solving quadratics by factoring, square root method, and quadratic formula.

⭐ Methods to Solve ax² + bx + c = 0

Factoring: set each factor = 0 (Zero Product Property)
Square Root: x² = k → x = ±√k (use when b = 0)
Quadratic Formula: x = (−b ± √(b²−4ac)) / (2a)
Discriminant: b²−4ac > 0 → 2 real solutions; = 0 → 1 solution; < 0 → no real solutions

x = (−b ± √(b² − 4ac)) / (2a)

✏️ Example

Solve: x² − 5x + 6 = 0

Factor: need two numbers that multiply to 6 and add to −5

(−2)(−3) = 6 ✓ and (−2)+(−3) = −5 ✓

So: (x − 2)(x − 3) = 0

Zero Product: x − 2 = 0 → x = 2, or x − 3 = 0 → x = 3

✅ Answer: x = 2 or x = 3

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"Dino's secret: ALWAYS check your answers by substituting back into the original equation! Even I make mistakes sometimes! 🦖"

🦕 Algebra 1 with Dino!

🎯 Dino's Big Quiz!

📋 Answer Key & Explanations