Mathematics Summary: Chapters 1-9

Chapter 1: Introduction to Real Numbers and Algebraic Expressions

Definitions

Algebraic Expression

A combination of variables, numerals (numbers), and operation signs to represent a value.

Example: $5a + 6b - 2$

Variable

A symbol (usually a letter) representing an unknown value.

Let $a =$ your age.

Absolute Value ($|x|$)

The distance of a number from zero on a number line. It is always non-negative.

$|-3| = 3$ and $|-0.75| = 0.75$

Sets of Numbers Hierarchy

Nested subsets of the real numbers:

  • Natural Numbers ($\mathbb{N}$): $\{1, 2, 3, ...\}$
  • Whole Numbers ($\mathbb{W}$): $\{0, 1, 2, 3, ...\}$
  • Integers ($\mathbb{Z}$): $\{..., -2, -1, 0, 1, 2, ...\}$
  • Rational Numbers ($\mathbb{Q}$): Numbers expressible as $\frac{a}{b}$
    • Examples: $\frac{5}{3}, 0.63, 0.0\overline{12}$
  • Irrational Numbers ($\mathbb{I}$): Non-terminating, non-repeating decimals
    • Examples: $\sqrt{3}, \pi, 0.10100110...$

Rules and Properties

Algebraic Laws

  • Commutative Laws: Order does not affect sums or products

    $a + b = b + a$, $ab = ba$

  • Associative Laws: Regrouping addends or factors leaves the result unchanged

    $(a + b) + c = a + (b + c)$, $(ab)c = a(bc)$

  • Distributive Law: $a(b + c) = ab + ac$

    $7(x - y) = 7x - 7y$

    $-5(x - 3y + 2z) = -5x + 15y - 10z$

Identity and Special Properties

  • Identity Properties: Zero is additive identity and one is multiplicative identity

    $a + 0 = a$ and $a \cdot 1 = a$

  • Properties of Reciprocals: The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ and keeps the same sign as the original number
  • Division Involving Zero: Zero divided by a non-zero number is zero, but dividing by zero is undefined

    $\frac{0}{a} = 0$ (for $a \neq 0$) and $\frac{a}{0}$ is undefined

Operations with Real Numbers

  • Addition of Real Numbers:
    • Different Signs: Subtract the smaller absolute value from the larger; keep the sign of the number with the larger absolute value

    $12 + (-9) = 3$ (Since 12 is larger than 9, the answer is positive)

    $-8 + 5 = -3$ (Since 8 is larger than 5, the answer is negative)

  • Subtraction Rule: Subtracting a number is the same as adding its opposite

    $3 - 7 = 3 + (-7) = -4$

    $-4 - (-10) = -4 + 10 = 6$

Other Rules

  • Order of Operations: Follow the order: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

    $5(3+4) - \{8 - [5 - (9+6)]\} = 17$

  • Translating Phrases to Expressions: Converting worded problems into mathematical symbols

    "Six more than some number" translates to $x + 6$

    "Five less than the product of two numbers" translates to $xy - 5$

Chapter 2: Solving Equations and Inequalities

Definitions

Solution

A replacement for the variable that makes the equation true.

For $x + 12 = 21$, if we replace $x$ with $9$, it is true.

Linear Inequality

A number sentence containing $>$, $<$, $\geq$, or $\leq$.

$x < 3$ (Numbers less than 3)

Translation Keywords Table

Vocabulary cues for operations:

  • Addition: "increased by", "added to"
  • Division: "ratio of", "quotient of", "per"

Rules

Equation Solving Principles

  • Addition Principle: You can add or subtract the same number from both sides of an equation to solve it

    Solve $y - 17.9 = -8.3$ by adding $17.9$ to both sides

  • Multiplication Principle: You can multiply or divide both sides by the same non-zero number

    Solve $-7x = 84$ by dividing both sides by $-7$

  • Solving Multi-Step Equations: Combine like terms and isolate the variable

    $4x + 7 - 6x = 10 + 3x + 12$ simplifies to $-2x + 7 = 3x + 22$

    Solving yields $x = -3$

Inequality Rules

  • Inequality Rule (Negative Multiplication): When multiplying or dividing by a negative number, reverse the inequality symbol

    $-4y < 20$ becomes $y > -5$ (Dividing by $-4$ reverses $<$ to $>$)

  • Inequality Translation Keywords:
    • "At least" indicates $\geq$
    • "At most" indicates $\leq$

    "Brian is at least 16" $\Rightarrow b \geq 16$

    "At most 3 students failed" $\Rightarrow s \leq 3$

Special Cases and Formulas

  • Special Case Solutions: Identities lead to infinite solutions and contradictions lead to none

    $6x - 15 = 3(2x - 5) \Rightarrow 0 = 0$ (infinite solutions)

    $2x + 5 = 2x - 15 \Rightarrow 5 = -15$ (no solution)

  • Formulas: Solving for a specific variable in a relationship

    Solve $C = \pi d$ for $d$. Result: $d = \frac{C}{\pi}$

  • Percent Formula: Translating "What is $P$% of $B$?" to an equation

    "What is 25% of 60?" translates to $a = 0.25 \cdot 60$

Chapter 3: Graphs of Linear Equations

Definitions

Quadrants

The four regions of the coordinate plane, numbered counterclockwise starting from the upper right:

  • Quadrant I: Both x and y are positive $(+, +)$
  • Quadrant II: x is negative, y is positive $(-, +)$
  • Quadrant III: Both x and y are negative $(-, -)$
  • Quadrant IV: x is positive, y is negative $(+, -)$

Note: Points on the x-axis or y-axis are not in any quadrant.

Point $(-4, 5)$ is in Quadrant II

Point $(3, 0)$ is on the x-axis (not in a quadrant)

Linear Equation

An equation whose graph is a straight line.

Example: $y = 3x$

Intercepts

Where the line crosses the axes.

  • y-intercept: Let $x=0$

    For $5x + 2y = 10$, if $x=0$, $2y=10 \rightarrow y=5$

    Point: $(0, 5)$

  • x-intercept: Let $y=0$

    For $5x + 2y = 10$, if $y=0$, $5x=10 \rightarrow x=2$

    Point: $(2, 0)$

Rules

Quadrant Sign Patterns

  • Quadrant I $(+, +)$
  • Quadrant II $(-, +)$
  • Quadrant III $(-, -)$
  • Quadrant IV $(+, -)$

Slope Formula

The ratio of vertical change to horizontal change.

For points $(-4, 5)$ and $(4, -1)$:

Slope $m = \frac{-1 - 5}{4 - (-4)} = \frac{-6}{8} = -\frac{3}{4}$

Horizontal Line

Equation $y = b$. The slope is 0.

$y = 2$ is a horizontal line passing through $(0, 2)$

Vertical Line

Equation $x = a$. The slope is undefined.

$x = -2$ is a vertical line passing through $(-2, 0)$

Chapter 4: Polynomials and Operations

Definitions

Polynomial

A monomial or sum/difference of monomials.

Example: $5x^4 - 8x^2y + y - 9$

Rules for Polynomials

Permit only whole-number exponents, no variables in denominators or radicals, and real-number coefficients.

Valid Polynomials:

  • $3x^2 - 5x + 7$ ✓
  • $-2y^3 + y^2 - 4$ ✓
  • $x^4y^2 - 3xy + 8$ ✓

NOT Polynomials (Common Mistakes):

  • $\frac{3}{x} + 2x$ ✗ (variable in denominator)
  • $x^{-2} + 5$ ✗ (negative exponent)
  • $\sqrt{x} + 3$ ✗ (same as $x^{1/2} + 3$, fractional exponent)
  • $x^{2.5} - 7$ ✗ (non-whole number exponent)
  • $5x^2 - \frac{2}{x^3}$ ✗ (same as $5x^2 - 2x^{-3}$, negative exponent)

Types of Polynomials

Classified by term count:

  • Monomial (one term): $7x^3$, $-5$, $2xy^2$
  • Binomial (two terms): $x^2 - 9$, $3a + 5b$
  • Trinomial (three terms): $x^2 + 5x + 6$, $a^2 - 2ab + b^2$

Polynomial Ordering

Standard form lists terms in descending exponent order; ascending order reverses that.

Standard Form (Descending):

$5x^4 - 3x^3 + 7x^2 - 2x + 9$

Ascending Form:

$9 - 2x + 7x^2 - 3x^3 + 5x^4$

Common Mistake:

Writing $3x + 7x^3 - 2x^2 + 5$ (not in order)

Degree of a Polynomial

The highest degree of any of its terms.

In $4x^2 - 9x^3 + 6x^4 + 8x - 7$, the highest exponent is 4, so the degree is 4.

Tricky Examples:

  • $3x^2y^3 + 5xy - 2$: Degree is 5 (from $x^2y^3$, add exponents: $2+3=5$)
  • $7x^4 - 2x^4 + x^2$: Degree is 4 (even though $x^4$ terms combine)
  • $-8$: Degree is 0 (constant term is $-8x^0$)
  • $5x$: Degree is 1 (same as $5x^1$)

Scientific Notation

Writing numbers as $M \times 10^n$.

$0.000567 = 5.67 \times 10^{-4}$

Rules (Exponents)

Basic Exponent Rules

  • Product Rule: Add exponents when multiplying like bases

    $a^m \cdot a^n = a^{m+n}$

    $m^5 \cdot m^2 = m^{5+2} = m^7$

  • Quotient Rule: Subtract exponents when dividing like bases

    $\frac{m^5}{m^2} = m^{5-2} = m^3$

  • Power Rule: Multiply exponents when raising a power to a power

    $(y^6)^{-3} = y^{-18}$

  • Zero and Negative Exponents:

    $a^0 = 1$ (for $a \neq 0$) and $a^{-n} = \frac{1}{a^n}$

    $(-5)^0 = 1$ and $3^{-4} = \frac{1}{81}$

Scientific Notation

Scientific Notation Constraint: The coefficient $M$ must satisfy $1 \leq M < 10$

$6.8 \times 10^5$ for 680000

Operations on Polynomials

Basic Operations

  • Addition: Combine like terms

    $(-6x^3 + 7x - 2) + (5x^3 + 4x^2 + 3) = -x^3 + 4x^2 + 7x + 1$

  • Dividing a Polynomial by a Monomial: Divide each term separately before simplifying

    $(21a^5b^4 - 14a^3b^2 + 7a^2b) \div (-7a^2b) = -3a^3b^3 + 2ab - 1$

Multiplication Rules

  • Multiplication (FOIL): For two binomials, multiply First, Outer, Inner, Last terms

    $(x + 4)(x^2 + 3) = x^3 + 3x + 4x^2 + 12$

  • Difference of Squares:

    $(A+B)(A-B) = A^2 - B^2$

    $(x + 8)(x - 8) = x^2 - 64$

  • Square of Binomial:

    $(A+B)^2 = A^2 + 2AB + B^2$

    $(x + 8)^2 = x^2 + 16x + 64$

    If $x^2 + y^2 = 20$ and $xy = 10$, then $(x + y)^2 = 40$

    Proof: $(x + y)^2 = x^2 + 2xy + y^2 = (x^2 + y^2) + 2xy = 20 + 2(10) = 40$

Chapter 5: Polynomials: Factoring

Definitions

GCF (Greatest Common Factor)

The largest factor common to all terms.

For $30x^3, -48x^4, 54x^5, 12x^2$, the GCF is $6x^2$

Prime Polynomial

A polynomial that cannot be factored further.

$x^2 - x + 7$

Rules and Methods

Start by diagnosing the expression: look for a GCF first, then choose a method based on term count (two terms → special products, three terms → trinomial methods, four or more → grouping or hybrids).

1. Factoring GCF

Always the first step.

$28x^6 + 32x^3 = 4x^3(7x^3 + 8)$

2. Factoring by Grouping (4 terms)

Group terms, factor GCF from each group, then factor the common binomial.

$3x^3 + 9x^2 + x + 3 = 3x^2(x+3) + 1(x+3) = (3x^2+1)(x+3)$

3. Difference of Squares

$A^2 - B^2 = (A+B)(A-B)$

$x^2 - 9 = (x+3)(x-3)$

Sum of Squares Reminder: $A^2 + B^2$ is prime over the reals

$16x^2 + 25$ (prime)

4. Factoring Trinomials ($x^2 + bx + c$)

Find two numbers that multiply to $c$ and add to $b$.

$y^2 - 8y + 15$

Factors of 15 that add to -8 are -3 and -5

Result: $(y-3)(y-5)$

5. ac-Method ($ax^2 + bx + c$)

Multiply $a \cdot c$, find factors of $ac$ that add to $b$, split the middle term.

$4x^2 - 5x - 6$

Multiply $4(-6) = -24$

Factors of -24 adding to -5 are -8 and 3

Split: $4x^2 - 8x + 3x - 6$, then group to solve

6. Zero Product Principle

If $(x-a)(x-b) = 0$, then $x=a$ or $x=b$.

$(x+4)(x-3) = 0 \Rightarrow x = -4$ or $x = 3$

7. Perfect Square Trinomials

First and last terms are squares and the middle term is twice their product, so factor to a binomial square.

$25x^2 - 20x + 4 = (5x - 2)^2$

Steps to Factor Polynomials

Factoring Decision Tree

1. GCF? → Factor it out first (ALWAYS!)

2. Count terms:

  • 2 terms? → Difference of Squares
  • 3 terms? → Perfect Square? → a = 1? → Simple / AC Method
  • 4+ terms? → Grouping

3. Equation? → Factor, then solve

2 TERMS

Difference of Squares

  • Pattern: $A^2 - B^2 = (A + B)(A - B)$
  • Check: Both terms perfect squares? Subtraction?

$9x^2 - 25 = (3x + 5)(3x - 5)$

3 TERMS

1: Perfect Square Trinomial

  • Pattern: $A^2 \pm 2AB + B^2 = (A \pm B)^2$
  • Test: Middle = $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$?

$4x^2 + 20x + 25 = (2x + 5)^2$

2: Simple Trinomial (a = 1)

  • Form: $x^2 + bx + c$
  • Find: Two numbers that multiply to c and add to b

$x^2 + 7x + 12 = (x + 3)(x + 4)$

3: AC Method (a ≠ 1)

  • Form: $ax^2 + bx + c$
  • Steps:
    1. Multiply a × c
    2. Find two numbers: multiply to ac, add to b
    3. Split middle term with these numbers
    4. Group and factor

Example: $3x^2 + 10x + 8$

  • $3 \times 8 = 24$
  • $4 \times 6 = 24$, $4 + 6 = 10$ ✓
  • $3x^2 + 4x + 6x + 8$
  • $(3x + 4)(x + 2)$

Chapter 6: Rational Expressions and Equations

Definitions

Excluded Values

Values that make the denominator zero.

In $\frac{x+3}{x^2 - 3x - 28}$, the denominator factors to $(x-7)(x+4)$

Excluded values are $7$ and $-4$

Rules

Simplifying

Factor numerator and denominator, then cancel common factors.

$\frac{3x - 12}{5x - 20} = \frac{3(x-4)}{5(x-4)} = \frac{3}{5}$

Multiplication

Multiply straight across.

$\frac{x^2 + 5x + 4}{x^2 - 9} \cdot \frac{x - 3}{x + 4}$ (After factoring and canceling, this simplifies)

Division

Multiply by the reciprocal.

$\frac{x}{9} \div \frac{8}{y} = \frac{x}{9} \cdot \frac{y}{8} = \frac{xy}{72}$

Addition/Subtraction

Must have a common denominator (LCD).

$\frac{4x}{x-7} + \frac{3x+2}{x-7} = \frac{7x+2}{x-7}$

Solving Rational Equations

Multiply the whole equation by the LCD to clear fractions.

$\frac{x}{5} = \frac{x}{2} - \frac{1}{4}$

Multiply by 20: $4x = 10x - 5$

Solving Proportions

Cross-multiply to solve $\frac{A}{B} = \frac{C}{D}$.

$\frac{x-2}{x-3} = \frac{x-1}{x+1}$

$(x-2)(x+1) = (x-1)(x-3)$ leading to $x = \frac{5}{3}$

Chapter 7: Graphs, Functions, and Applications

Definitions

Function

A relationship where each input (domain) has exactly one output (range).

Domain $\{-5, 1, 3, 4\}$, Range $\{1, 0, -5, 3\}$ from graph points

Relation vs. Function

Relations can map one input to multiple outputs, but functions cannot.

Domain $\{9\}$ mapping to range $\{-3, 3\}$ is a relation, not a function

Vertical Line Test

If a vertical line touches the graph more than once, it is not a function.

A sideways parabola is not a function because a vertical line crosses it twice

Specific Domain Rules

  • Polynomial/linear/quadratic functions have domain $(-\infty, \infty)$
  • Rational functions exclude denominator zeros

    $f(x) = \frac{3x}{x-4}$ has domain $\{x | x \neq 4\}$

  • Constant functions have domain $(-\infty, \infty)$ but range $\{c\}$

Rules regarding Lines

Slope-Intercept Form

$y = mx + b$

$f(x) = -3.1x + 7$

Slope $m = -3.1$, y-intercept is $(0, 7)$

Point-Slope Form

$y - y_1 = m(x - x_1)$

Line with slope 3 through $(2, 7)$ becomes $y - 7 = 3(x - 2)$

Simplifying to $y = 3x + 1$

Parallel Lines

Have the same slope.

$y = -2x - 3$ is parallel to $8x + 4y = -6$

(which simplifies to $y = -2x - 1.5$)

Both slopes are -2

Perpendicular Lines

Slopes are negative reciprocals (product is -1).

$y = 4x + 1$ is perpendicular to $x + 4y = 4$ (slope $-1/4$)

Chapter 8: Systems of Equations

Solving Methods

1. Graphing

Find the intersection point of two lines.

Graphing $y = 3x + 1$ and $x - 2y = 3$ reveals the intersection point $(-1, -2)$

2. Substitution

Solve one equation for a variable and plug it into the other.

System: $x = 3 - y$ and $5x + 3y = 5$

Substitute $(3-y)$ for $x$ in the second equation:

$5(3-y) + 3y = 5$

3. Elimination

Add equations to eliminate a variable.

$4x + 3y = 8$

$x - 3y = 7$

Adding these gives $5x = 15$, so $x = 3$

Classifications

Inconsistent

Lines are parallel (No solution).

$y = \frac{3}{4}x + 2$ and $y = \frac{3}{4}x - 3$

Dependent

Lines are identical (Infinite solutions).

$4x + 8y = 16$ and $2x + 4y = 8$

Consistent System

Has at least one solution (lines intersect).

Independent Equations

Represent distinct lines.

Classification Summary

  • One solution → Consistent and Independent
  • No solution → Inconsistent and Independent (parallel)
  • Infinite solutions → Consistent and Dependent (same line)

Chapter 9: More on Inequalities

Definitions

Intersection ($\cap$)

Elements common to both sets ("AND").

Intersection of $\{a, b, c, d, e\}$ and $\{a, e, i, o, u\}$ is $\{a, e\}$

Union ($\cup$)

Elements in either set ("OR").

Union of the sets above is $\{a, b, c, d, e, i, o, u\}$

Rules

Compound Inequalities

$6 < 2x - 4 < 10$

Add 4 to all parts: $10 < 2x < 14$

Divide by 2: $5 < x < 7$

Absolute Value Equations

$|X| = p$ means $X = p$ or $X = -p$.

$|x| = 7 \Rightarrow x = 7$ or $x = -7$

$|2x + 1| = 5 \Rightarrow 2x+1=5$ or $2x+1=-5$

Absolute Value Inequalities

  • $|X| < p$ (Less than means "Between"):

    $|x| < 3 \Rightarrow -3 < x < 3$

  • $|X| > p$ (Greater than means "Split"):

    $|x| \geq 3 \Rightarrow x \geq 3$ or $x \leq -3$

Special Cases for Absolute Value Inequalities

  • $|x| < -p$ has no solution because absolute values are non-negative ($\emptyset$)
  • $|x| > -p$ is true for all real numbers since absolute values exceed any negative bound ($(-\infty, \infty)$)

Graphing Inequalities (2 Variables)

Key Rules:

  • Use dashed line for $<$ or $>$ (boundary NOT included)
  • Use solid line for $\leq$ or $\geq$ (boundary included)
  • Test a point to determine which side to shade

Example 1: To graph $y < \frac{1}{2}x - 1$:

  1. Draw the line $y = \frac{1}{2}x - 1$ as a dashed line (because of $<$)
  2. Test point $(0,0)$: $0 < -1$ is False
  3. Shade the region below the line

Example 2: To graph $y \geq 2x + 3$:

  1. Draw the line $y = 2x + 3$ as a solid line (because of $\geq$)
  2. Test point $(0,0)$: $0 \geq 3$ is False
  3. Shade the region above the line

Note: Points on the solid line are part of the solution!