Study Guide
Core Concepts & Formulas
Unit 1
Variables, Expressions & Order of Operations
A variable is a letter representing an unknown value. An algebraic expression combines variables, numbers, and operations. The order of operations (PEMDAS) determines the sequence for evaluating expressions.
📌 Memorize
PEMDAS: Parentheses → Exponents → Multiply/Divide (L→R) → Add/Subtract (L→R)
Like terms: same variable & same exponent (e.g., 3x and 7x are like terms)
Distributive Property: a(b + c) = ab + ac
📐 Example
Simplify: $3(2x - 4) + 5x$
Distribute: $6x - 12 + 5x$
Combine like terms: $11x - 12$ ✓
Unit 2
Solving Linear Equations
A linear equation is an equation where the variable has a power of 1. Solve by performing inverse operations to isolate the variable. Always check your answer by substituting back.
📌 Memorize
Goal: isolate the variable on one side
Whatever you do to one side, do to the other
Equation with no solution: contradiction (e.g., 3 = 5); Infinite solutions: identity (e.g., x = x)
📐 Example
Solve: $\frac{2x+3}{5} = 7$
Multiply both sides by 5: $2x + 3 = 35$
Subtract 3: $2x = 32$
Divide by 2: $x = 16$ ✓
Unit 3
Linear Inequalities
Solve inequalities like equations, except: when multiplying or dividing by a negative number, flip the inequality sign. Graph solutions on a number line: open circle for </>, closed circle for ≤/≥.
📌 Memorize
Multiply/divide by a negative → FLIP the sign!
$|ax + b| < c$ → $-c < ax + b < c$ (AND)
$|ax + b| > c$ → $ax + b < -c$ OR $ax + b > c$
📐 Example
Solve: $-3x + 2 \geq 14$
Subtract 2: $-3x \geq 12$
Divide by -3 (FLIP!): $x \leq -4$ ✓
Unit 4
Functions & Relations
A function is a relation where each input (x) has exactly one output (y). Use the vertical line test on a graph. Linear functions: $f(x) = mx + b$.
📌 Memorize
Domain = all valid inputs (x-values)
Range = all possible outputs (y-values)
Function notation: $f(3)$ means substitute $x = 3$
Rate of change = slope = $\frac{\Delta y}{\Delta x}$
📐 Example
If $f(x) = 3x^2 - 2x + 1$, find $f(-1)$
$f(-1) = 3(-1)^2 - 2(-1) + 1 = 3 + 2 + 1 = 6$ ✓
Unit 5
Graphing Linear Functions
The slope-intercept form is $y = mx + b$, where $m$ = slope and $b$ = y-intercept. The point-slope form is $y - y_1 = m(x - x_1)$. Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes.
📌 Memorize
Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Parallel: $m_1 = m_2$; Perpendicular: $m_1 \cdot m_2 = -1$
Horizontal line: $y = k$ (slope = 0); Vertical line: $x = k$ (undefined slope)
📐 Example
Find slope of the line through $(2, 5)$ and $(-1, -4)$
$m = \frac{-4 - 5}{-1 - 2} = \frac{-9}{-3} = 3$ ✓
Unit 6
Systems of Linear Equations
A system has two or more equations with the same variables. Solve by: substitution, elimination, or graphing. The solution is the ordered pair $(x, y)$ that satisfies all equations simultaneously.
📌 Memorize
One solution: lines intersect (consistent & independent)
No solution: parallel lines (inconsistent)
Infinite solutions: same line (consistent & dependent)
📐 Example
Solve: $x + y = 7$ and $x - y = 3$
Add equations: $2x = 10 \Rightarrow x = 5$
Substitute: $5 + y = 7 \Rightarrow y = 2$
Solution: $(5, 2)$ ✓
Unit 7
Exponent Rules & Polynomials
Polynomials are expressions with one or more terms. Learn exponent rules to simplify. Adding/subtracting polynomials: combine like terms. Multiplying: use distributive property or FOIL for binomials.
📌 Memorize
Product rule: $x^a \cdot x^b = x^{a+b}$
Quotient rule: $x^a \div x^b = x^{a-b}$
Power rule: $(x^a)^b = x^{ab}$
Zero exponent: $x^0 = 1$ ($x \neq 0$)
Negative exponent: $x^{-n} = \frac{1}{x^n}$
FOIL: $(a+b)(c+d) = ac + ad + bc + bd$
📐 Example
Expand: $(2x + 3)(x - 5)$
FOIL: $2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15$ ✓
Unit 8
Factoring Polynomials
Factoring is the reverse of expanding. Always check for the GCF first. Then identify: difference of squares, perfect square trinomials, or factor trinomials by finding two numbers that multiply to $ac$ and add to $b$.
📌 Memorize
GCF first! Always!
Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
Perfect square: $a^2 + 2ab + b^2 = (a+b)^2$
Trinomial $x^2 + bx + c$: find two numbers that multiply to $c$, add to $b$
📐 Example
Factor: $x^2 - 5x + 6$
Need two numbers: $(-2) \times (-3) = 6$, $(-2)+(-3) = -5$
$x^2 - 5x + 6 = (x-2)(x-3)$ ✓
Unit 9
Quadratic Equations
A quadratic equation has the form $ax^2 + bx + c = 0$. Solve by: factoring, completing the square, or the quadratic formula. The discriminant $b^2 - 4ac$ tells you the number of real solutions.
📌 Memorize
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $b^2-4ac > 0$ → 2 real solutions
Discriminant: $b^2-4ac = 0$ → 1 real solution
Discriminant: $b^2-4ac < 0$ → no real solutions
📐 Example
Solve: $x^2 - 5x + 6 = 0$
Factor: $(x-2)(x-3) = 0$
$x = 2$ or $x = 3$ ✓
Unit 10
Radical Expressions & Data Analysis
Radicals: $\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}$. Simplify by finding perfect square factors. In statistics, understand mean, median, mode, range, and how outliers affect measures of central tendency.
📌 Memorize
$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$; $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
Mean = sum ÷ count; Median = middle value
Range = max − min; Mode = most frequent
Outlier affects mean most, not median
📐 Example
Simplify: $\sqrt{72}$
$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$ ✓
Exam · 20 Problems
Practice Problems
Instructions
Answer each question in the input box and click Check. You'll get immediate feedback. Submit all answers at the end to see your final score. Write simplified, exact answers (e.g., write fractions in lowest terms, radicals in simplest form). Correct answers earn 5 points each.