🦕 Dino Algebra 1 — Complete Review

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Unit 1: Real Numbers & Properties

Number sets, order of operations, properties of real numbers

📖 Key Concepts

Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real
Order of Operations: PEMDAS (Parentheses → Exponents → Multiply/Divide → Add/Subtract)
Commutative: a+b=b+a | Associative: (a+b)+c=a+(b+c) | Distributive: a(b+c)=ab+ac
Absolute value: |x| = distance from zero, always ≥ 0

⚡ Must Memorize

PEMDAS: Please Excuse My Dear Aunt Sally

a(b + c) = ab + ac [Distributive Property]

|−x| = |x| = x, when x ≥ 0

✏️ Worked Example

Simplify: 3 + 2 × (4² − 6) ÷ 5

Step 1: Exponent first → 4² = 16
Step 2: Parentheses → 16 − 6 = 10
Step 3: Multiply → 2 × 10 = 20
Step 4: Divide → 20 ÷ 5 = 4
Step 5: Add → 3 + 4 = 7

🦕

Always tackle the exponents before multiplying! 🌟

1

Real Numbers

⭐ Easy

Which set of numbers is closed under division (for nonzero divisors)?

2

Order of Operations

🔥 Medium

Evaluate: 18 ÷ 2 + (3² − 4) × 2

⚖️

Unit 2: Solving Linear Equations

One-step, two-step, multi-step, literal equations

📖 Key Concepts

Solve by isolating the variable — use inverse operations
Whatever you do to one side, do to the other
Literal equation: solve for one variable in terms of others
No solution: variables cancel, false statement (e.g., 3 = 5)
Infinite solutions: variables cancel, true statement (e.g., 5 = 5)

⚡ Must Memorize

ax + b = c → x = (c − b) / a

Combine like terms FIRST, then isolate variable

✏️ Worked Example

Solve: 3(2x − 4) = 2x + 8

6x − 12 = 2x + 8 [distribute]
4x = 20 [subtract 2x, add 12]
x = 5

🦖

Distribute first, then collect like terms! 🦴

3

Linear Equations

🔥 Medium

Solve for x: 5x − 3 = 2x + 12

4

Literal Equations

💀 Hard

Solve for h: A = ½bh

📏

Unit 3: Inequalities

Linear inequalities, compound inequalities, graphing on a number line

📖 Key Concepts

Solve like equations, BUT flip the inequality sign when multiplying or dividing by a NEGATIVE number
AND (∩): both conditions must be true — shaded region overlaps
OR (∪): at least one condition true — two separate rays
Open circle ○ = not included; Closed circle ● = included

⚡ Must Memorize

Multiply / Divide by NEGATIVE → FLIP the sign!

−3x > 9 → x < −3 ✔

✏️ Worked Example

Solve: −4x + 3 ≤ 15

−4x ≤ 12 [subtract 3]
x ≥ −3 [divide by −4, FLIP sign]
Answer: x ≥ −3

🦕

Dividing by a negative? Don't forget to flip! 🔄

5

Inequalities

🔥 Medium

Solve: −2x + 7 > 13

📈

Unit 4: Functions & Relations

Domain, range, vertical line test, function notation

📖 Key Concepts

Function: each input (x) maps to exactly ONE output (y)
Vertical Line Test: if any vertical line hits the graph twice → NOT a function
Domain = all valid x-inputs; Range = all resulting y-outputs
f(a) means substitute x = a into the function

⚡ Must Memorize

f(x): one x → one y [function rule]

Domain: x-values that work (no division by 0, no √negative)

✏️ Worked Example

If f(x) = 2x² − 3x + 1, find f(−2)

f(−2) = 2(−2)² − 3(−2) + 1
= 2(4) + 6 + 1 = 8 + 6 + 1 = 15

🦖

One input, one output — that's what makes a function! 🎯

6

Functions

🔥 Medium

If f(x) = 3x − 5, what is f(4)?

📉

Unit 5: Linear Functions & Graphing

Slope, slope-intercept form, standard form, parallel & perpendicular

📖 Key Concepts

Slope = rise/run = (y₂ − y₁)/(x₂ − x₁)
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; Perpendicular: slopes are negative reciprocals

⚡ Must Memorize

m = (y₂ − y₁) / (x₂ − x₁) [slope formula]

Parallel: m₁ = m₂ | Perpendicular: m₁ × m₂ = −1

✏️ Worked Example

Find the slope between (1, 3) and (4, 9)

m = (9 − 3) / (4 − 1) = 6 / 3 = 2

7

Slope & Graphing

🔥 Medium

What is the slope of the line passing through (−2, 5) and (4, −1)?

8

Linear Equations in Two Variables

💀 Hard

Which equation passes through (2, 1) and is perpendicular to y = 2x + 3?

🔀

Unit 6: Systems of Equations

Substitution, elimination, graphing, special systems

📖 Key Concepts

Substitution: isolate one variable, substitute into other equation
Elimination: add/subtract equations to cancel one variable
One solution: lines intersect at one point
No solution: parallel lines (same slope, different y-intercept)
Infinite solutions: same line (identical equations)

⚡ Must Memorize

Elimination: multiply equations to make coefficients equal, then add or subtract

✏️ Worked Example

Solve: x + y = 7 and x − y = 3

Add equations: 2x = 10 → x = 5
Substitute: 5 + y = 7 → y = 2
Answer: (5, 2)

🦕

Add the equations to eliminate a variable — magic! ✨

9

Systems of Equations

💀 Hard

Solve the system: 2x + 3y = 12 and x − y = 1

📦

Unit 7: Exponent Rules & Polynomials

Laws of exponents, adding, subtracting, multiplying polynomials

📖 Key Concepts

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

⚡ Must Memorize

xᵃ · xᵇ = xᵃ⁺ᵇ | (xᵃ)ᵇ = xᵃᵇ | x⁻ⁿ = 1/xⁿ

(a + b)² = a² + 2ab + b² [Perfect Square Trinomial]

(a + b)(a − b) = a² − b² [Difference of Squares]

✏️ Worked Example

Multiply: (x + 3)(x − 5)

FOIL: x·x + x·(−5) + 3·x + 3·(−5)
= x² − 5x + 3x − 15
= x² − 2x − 15

10

Exponents

🔥 Medium

Simplify: (3x²y³)² · (x⁻¹)

11

Polynomials — FOIL

🔥 Medium

Expand: (2x − 3)²

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Unit 8: Factoring Polynomials

GCF, trinomials, difference of squares, factoring by grouping

📖 Key Concepts

Always factor out the GCF first
Factor x² + bx + c: find two numbers that multiply to c and add to b
Difference of Squares: a² − b² = (a + b)(a − b)
Perfect Square Trinomial: a² ± 2ab + b² = (a ± b)²

⚡ Must Memorize

a² − b² = (a + b)(a − b)

x² + bx + c = (x + p)(x + q) where p·q=c and p+q=b

✏️ Worked Example

Factor: x² + 7x + 12

Find two numbers: multiply to 12, add to 7 → 3 and 4
Answer: (x + 3)(x + 4)

🦖

Always pull out the GCF before anything else! 🌿

12

Factoring

🔥 Medium

Factor completely: x² − 5x − 14

13

Difference of Squares

⭐ Easy

Factor: 4x² − 49

🎯

Unit 9: Quadratic Equations

Solving by factoring, square roots, completing the square, quadratic formula

📖 Key Concepts

Standard form: ax² + bx + c = 0
Solve by factoring → set each factor = 0 (Zero Product Property)
Quadratic Formula: x = [−b ± √(b²−4ac)] / 2a
Discriminant b²−4ac: >0 two real, =0 one real, <0 no real solutions
Vertex form: y = a(x−h)² + k → vertex at (h, k)

⚡ Must Memorize

x = [−b ± √(b² − 4ac)] / (2a) [Quadratic Formula]

Discriminant: b²−4ac > 0 → 2 real roots; = 0 → 1 root; < 0 → no real roots

✏️ Worked Example

Solve: x² − 5x + 6 = 0

Factor: (x − 2)(x − 3) = 0
Set each factor to 0: x − 2 = 0 → x = 2; x − 3 = 0 → x = 3
Answer: x = 2 or x = 3

🦕

When factoring fails, the quadratic formula never lets you down! 🚀

14

Quadratic Equations

🔥 Medium

Solve by the quadratic formula: x² − 3x − 10 = 0

15

Discriminant

💀 Hard

How many real solutions does 2x² − 4x + 5 = 0 have?

√

Unit 10: Radicals, Rationals & Data

Simplifying radicals, rational expressions, statistics basics

📖 Key Concepts

Simplify √n: factor out perfect squares → √(4·3) = 2√3
√a · √b = √(ab) | √a / √b = √(a/b)
Rationalize denominator: multiply by √b/√b to eliminate radical
Rational expressions: factor then cancel common factors
Mean = sum ÷ count; Median = middle value; Mode = most frequent

⚡ Must Memorize

√(a²b) = a√b (a ≥ 0) [simplifying radicals]

Rationalize: 1/√3 = √3/3

✏️ Worked Example

Simplify: √75

√75 = √(25 · 3) = √25 · √3 = 5√3

🦖

Look for perfect square factors hiding inside the radical! 🔎

16

Radicals

⭐ Easy

Simplify: √(48)

17

Statistics

⭐ Easy

Find the mean of: 4, 7, 10, 3, 6

18

Rational Expressions

💀 Hard

Simplify: (x² − 9) / (x² − x − 6)

19

Word Problem — Linear

💀 Hard

A store sells notebooks for $3 each and pens for $1.50 each. A student buys 10 items total and spends $24. How many notebooks were bought?

20

Quadratic — Vertex Form

💀 Hard

What is the vertex of the parabola: y = 2(x − 3)² + 5 ?