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Cheatsheet: Algebra Foundations

1. Real Numbers and Properties

1.1 Number Sets

Number Set Definition and Examples
Natural Numbers (ℕ) Counting numbers: {1, 2, 3, 4, ...}
Whole Numbers (W) Natural numbers and zero: {0, 1, 2, 3, ...}
Integers (ℤ) Whole numbers and their negatives: {..., -2, -1, 0, 1, 2, ...}
Rational Numbers (ℚ) Numbers that can be written as a/b where a, b are integers and b ≠ 0; includes terminating and repeating decimals
Irrational Numbers Numbers that cannot be written as fractions; non-terminating, non-repeating decimals (√2, π, √3)
Real Numbers (ℝ) All rational and irrational numbers combined

1.2 Properties of Real Numbers

Property Definition
Commutative Property (Addition) a + b = b + a
Commutative Property (Multiplication) a × b = b × a
Associative Property (Addition) (a + b) + c = a + (b + c)
Associative Property (Multiplication) (a × b) × c = a × (b × c)
Distributive Property a(b + c) = ab + ac
Identity Property (Addition) a + 0 = a
Identity Property (Multiplication) a × 1 = a
Inverse Property (Addition) a + (-a) = 0
Inverse Property (Multiplication) a × (1/a) = 1, where a ≠ 0
Zero Property of Multiplication a × 0 = 0

2. Order of Operations

2.1 PEMDAS/GEMDAS

  • P: Parentheses (and other grouping symbols like brackets, braces)
  • E: Exponents (powers and roots)
  • M/D: Multiplication and Division (left to right)
  • A/S: Addition and Subtraction (left to right)

2.2 Key Rules

  • Operations at the same level are performed left to right
  • Multiplication does not always come before division
  • Addition does not always come before subtraction
  • Evaluate innermost grouping symbols first

3. Variables and Expressions

3.1 Terminology

Term Definition
Variable A letter or symbol that represents an unknown value
Constant A fixed numerical value
Coefficient The numerical factor of a term containing a variable (in 5x, 5 is the coefficient)
Term A single number, variable, or product of numbers and variables (3, x, 5xy)
Expression A mathematical phrase combining numbers, variables, and operations (3x + 5)
Equation A mathematical statement showing two expressions are equal (3x + 5 = 11)

3.2 Types of Expressions

Type Definition
Monomial Expression with one term (5x²)
Binomial Expression with two terms (3x + 7)
Trinomial Expression with three terms (x² + 5x + 6)
Polynomial Expression with one or more terms (general term)

3.3 Like Terms

  • Terms with identical variable parts (same variables raised to same powers)
  • Examples: 3x and 7x are like terms; 3x² and 7x are not like terms
  • Combine like terms by adding or subtracting coefficients
  • Example: 5x + 3x = 8x

4. Evaluating Expressions

4.1 Substitution Process

  • Replace each variable with its given value
  • Use parentheses around substituted values
  • Follow order of operations to simplify
  • Example: Evaluate 3x + 5 when x = 4 → 3(4) + 5 = 12 + 5 = 17

5. Simplifying Expressions

5.1 Steps for Simplification

  • Remove parentheses using distributive property
  • Combine like terms
  • Write in standard form (descending order of exponents)

5.2 Distributive Property Applications

  • a(b + c) = ab + ac
  • a(b - c) = ab - ac
  • -(a + b) = -a - b
  • -(a - b) = -a + b

6. Exponent Rules

6.1 Basic Exponent Laws

Rule Formula
Product of Powers a^m × a^n = a^(m+n)
Quotient of Powers a^m ÷ a^n = a^(m-n), where a ≠ 0
Power of a Power (a^m)^n = a^(mn)
Power of a Product (ab)^n = a^n × b^n
Power of a Quotient (a/b)^n = a^n/b^n, where b ≠ 0
Zero Exponent a^0 = 1, where a ≠ 0
Negative Exponent a^(-n) = 1/a^n, where a ≠ 0
Identity a^1 = a

7. Linear Equations in One Variable

7.1 Solving One-Step Equations

Type Method
Addition Equation (x + a = b) Subtract a from both sides
Subtraction Equation (x - a = b) Add a to both sides
Multiplication Equation (ax = b) Divide both sides by a
Division Equation (x/a = b) Multiply both sides by a

7.2 Solving Multi-Step Equations

  • Step 1: Simplify each side (distribute, combine like terms)
  • Step 2: Move variable terms to one side using addition/subtraction
  • Step 3: Move constant terms to the other side
  • Step 4: Divide or multiply to isolate the variable
  • Step 5: Check solution by substituting back into original equation

7.3 Equations with Variables on Both Sides

  • Collect all variable terms on one side
  • Collect all constants on the other side
  • Solve for the variable
  • Example: 3x + 5 = 2x + 9 → x + 5 = 9 → x = 4

7.4 Special Cases

Type Characteristic
Identity (Infinitely Many Solutions) Results in a true statement (5 = 5); all real numbers are solutions
Contradiction (No Solution) Results in a false statement (3 = 7); no values satisfy the equation

8. Literal Equations and Formulas

8.1 Definition

  • Equations with two or more variables
  • Solving for one variable in terms of the others

8.2 Common Formulas

Formula Description
P = 2l + 2w Perimeter of rectangle (l = length, w = width)
A = lw Area of rectangle
A = ½bh Area of triangle (b = base, h = height)
d = rt Distance formula (r = rate, t = time)
C = 2πr Circumference of circle (r = radius)
A = πr² Area of circle

8.3 Solving Literal Equations

  • Use same steps as solving regular equations
  • Isolate the specified variable
  • Example: Solve A = lw for w → w = A/l

9. Ratios, Rates, and Proportions

9.1 Ratios

  • Comparison of two quantities: a:b or a/b
  • Can be simplified like fractions
  • Example: 6:8 simplifies to 3:4

9.2 Rates

  • Ratio comparing quantities with different units
  • Unit rate: rate with denominator of 1
  • Example: 120 miles in 2 hours = 60 miles per hour

9.3 Proportions

  • Equation stating two ratios are equal: a/b = c/d
  • Cross-multiplication: ad = bc
  • Used to solve for unknown values
  • Example: 3/4 = x/12 → 4x = 36 → x = 9

10. Percent

10.1 Percent Basics

  • Percent means "per hundred" (represented by %)
  • Convert percent to decimal: divide by 100 (25% = 0.25)
  • Convert decimal to percent: multiply by 100 (0.6 = 60%)
  • Convert fraction to percent: divide, then multiply by 100

10.2 Percent Equation

  • Part = Percent × Whole
  • Percent = Part/Whole
  • Whole = Part/Percent

10.3 Percent Applications

Application Formula
Percent Increase ((New - Original)/Original) × 100%
Percent Decrease ((Original - New)/Original) × 100%
Markup/Discount New Price = Original ± (Percent × Original)
Sales Tax Total = Price + (Tax Rate × Price)

11. Absolute Value

11.1 Definition

  • Distance from zero on number line (always non-negative)
  • |a| = a if a ≥ 0; |a| = -a if a <>
  • Example: |5| = 5, |-5| = 5

11.2 Properties

  • |a| ≥ 0 for all real numbers a
  • |a| = |-a|
  • |ab| = |a| × |b|
  • |a/b| = |a|/|b|, where b ≠ 0

12. Inequalities

12.1 Inequality Symbols

Symbol Meaning
> Greater than
Less than
Greater than or equal to
Less than or equal to
Not equal to

12.2 Solving Linear Inequalities

  • Use same steps as solving equations
  • Key rule: When multiplying or dividing by a negative number, reverse the inequality sign
  • Example: -2x > 6 → x < -3="" (sign="">
  • Solution represented on number line or in interval notation

12.3 Graphing on Number Line

  • Open circle (○): < or=""> (value not included)
  • Closed circle (●): ≤ or ≥ (value included)
  • Shade to the right for > or ≥
  • Shade to the left for < or="">

12.4 Interval Notation

Inequality Interval Notation
x > a (a, ∞)
x ≥ a [a, ∞)
x <> (-∞, b)
x ≤ b (-∞, b]
a < x=""><> (a, b)
a ≤ x ≤ b [a, b]

13. Coordinate Plane

13.1 Components

  • x-axis: horizontal number line
  • y-axis: vertical number line
  • Origin: point (0, 0) where axes intersect
  • Ordered pair: (x, y) where x is horizontal position, y is vertical position

13.2 Quadrants

Quadrant Signs
I (+, +)
II (-, +)
III (-, -)
IV (+, -)

14. Relations and Functions

14.1 Definitions

Term Definition
Relation Set of ordered pairs (x, y)
Domain Set of all x-values (input values)
Range Set of all y-values (output values)
Function Relation where each input has exactly one output

14.2 Vertical Line Test

  • Graph represents a function if no vertical line intersects it more than once
  • Used to determine if a graph is a function

14.3 Function Notation

  • f(x) read as "f of x" or "the value of f at x"
  • f(x) represents the output when x is the input
  • Example: If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11

15. Linear Functions

15.1 Slope

  • Measure of steepness of a line
  • m = (y₂ - y₁)/(x₂ - x₁) = rise/run
  • Positive slope: line rises left to right
  • Negative slope: line falls left to right
  • Zero slope: horizontal line
  • Undefined slope: vertical line

15.2 Forms of Linear Equations

Form Equation
Slope-Intercept Form y = mx + b (m = slope, b = y-intercept)
Point-Slope Form y - y₁ = m(x - x₁) (m = slope, (x₁, y₁) = point on line)
Standard Form Ax + By = C (A, B, C are integers, A ≥ 0)

15.3 Intercepts

Intercept Definition
x-intercept Point where line crosses x-axis (y = 0)
y-intercept Point where line crosses y-axis (x = 0)

15.4 Special Lines

Type Equation
Horizontal Line y = k (slope = 0)
Vertical Line x = h (undefined slope)
Parallel Lines Same slope, different y-intercepts (m₁ = m₂)
Perpendicular Lines Slopes are negative reciprocals (m₁ × m₂ = -1)
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