Complete Algebra Masterclass 2026: Zero to Pre-Calculus

The vocabulary of algebra: what a variable is, what a term is, what a coefficient is, and why distinguishing them matters. We work through expressions of increasing complexity, identifying each piece. By the end you will talk about algebra precisely instead of pointing at "that thing with the x", and that precision is what makes the rest of the course click into place.

The two most-used moves in algebra: combining like terms (3x plus 5x equals 8x) and distributing a factor across a sum (2 times the quantity x plus 3 equals 2x plus 6). We work through both rules with single and multiple variables, identify which terms can and cannot be combined, and walk through the order in which to apply distribution and combination to simplify any expression cleanly.

Parentheses inside parentheses inside brackets, the kind of expression that makes students panic. We show the systematic inside-out approach that handles any level of nesting without confusion: identify the innermost group, simplify it, expand outward, and combine. After this lesson, even six-level-deep expressions become routine.

Substituting numbers into algebraic expressions and computing the result, a foundational skill used in every test, every word problem, and every real-world application. We cover substitution into single-variable and multi-variable expressions, expressions with exponents and absolute values, and the subtle traps with negative numbers (parenthesize, always).

The translation skill that unlocks every word problem. We work through the standard verbal patterns ("the sum of", "the difference of", "twice as many", "five less than") and convert each to algebra systematically. By the end, English sentences turn into equations almost automatically, a skill you will lean on heavily in Sections 3, 4, 6, 9, 11, and 13.

The common pitfalls in word-to-algebra translation that turn easy problems into wrong answers. "Five less than x" is x minus 5, not 5 minus x. "The difference of a and b" is a minus b, not the absolute value of (a minus b). "Twice the sum" is 2 times the quantity (a plus b), not 2a plus b. We hit every famous trap with worked examples so you recognize each one on a test before it traps you.

A cheat sheet of the algebraic expression shapes you will see thousands of times: perimeter formulas, area formulas, rate formulas, distance formulas. And how to recognize the underlying template behind any new word problem. This lesson is the bridge between translating one sentence and modeling an entire problem. The template-recognition skill is what test-prep books charge thousands of dollars to teach.

A full set of worked examples consolidating Section 2: simplifying, distributing, evaluating, translating, and recognizing templates. Use this lesson before any test or as a refresher before moving into the equation-solving sections that follow. Every problem worked step by step, with the most common student traps highlighted.

A contractor estimating materials for a remodeling job uses every Section 2 skill: setting up expressions for area and perimeter, evaluating with measured dimensions, and translating verbal client requests into algebra. We work through a real bathroom-renovation estimate from quote to final number. The same template-translation skill transfers directly to budgeting, project planning, and any field that turns words into numbers.

The definition of a linear equation, the goal of solving one (isolate the variable), and the two operations that get you there: same operation on both sides, and combining like terms. We walk through one-step and two-step linear equations and establish the playbook every later equation type extends.

Equations of the form ax plus b equals c, the workhorse of Algebra 1 and the building block for everything more complex. We solve dozens of variations: positive and negative coefficients, fractional coefficients, equations with parentheses, and the standard "show every step" format that prevents arithmetic errors. By the end you will solve any two-step equation in under 30 seconds.

When the variable appears on both sides of the equal sign (4x plus 7 equals 2x plus 15), many students freeze. We show the standard "move variables to one side first" routine, then finish with the two-step technique from the previous lesson. This pattern is the foundation of every later equation and inequality method, so we nail it down with multiple examples.

Solving for a specific variable in a formula like A equals pi r squared, F equals m a, or P V equals n R T. This skill is essential for science, engineering, and finance, where formulas are stated in one form but you need a different unknown. We work through 6 to 8 standard formulas, isolating each variable in turn, and the same algebraic techniques generalize to any literal equation you will meet later.

Inequalities are equations with a twist: the rule "flip the inequality when you multiply or divide by a negative". We solve linear inequalities, graph the solutions on the number line, and write them in interval notation. Every inequality skill in this lesson reappears in absolute value inequalities, polynomial inequalities, and systems of inequalities later in the course.

Compound inequalities have two conditions joined by AND or OR, for example "x greater than 2 AND x less than 7" written as 2 less than x less than 7. We cover both joining types, intersection vs union of solution sets, graphing on the number line, and the handful of trick cases (a "no solution" or "all real numbers" outcome). Compound inequalities are tested heavily on the SAT and ACT.

Solving absolute value of an expression equals a number requires splitting into two cases, the most-skipped step in algebra, and the most-tested. We cover the standard split, the "no solution" case (when the right side is negative), the "isolate first" preparation, and a worked tournament of examples that hit every flavor of absolute value equation you will see on a test.

The two-case rule for absolute value inequalities, with a key twist: absolute value of x less than a means negative a less than x less than a (an AND), and absolute value of x greater than a means x less than negative a OR x greater than a. Confusing these two cases is the number 1 algebra-test mistake, and this lesson nails the difference with the "less than" / "greater than" memory device that makes it permanent.

A full set of word problems requiring linear equations and inequalities: age problems, mixture problems, motion problems, work-rate problems, and consecutive integer problems. We use the Words-to-Algebra technique from Section 2 systematically, translating each problem step by step, then solving the equation that results. These are the problems that earn full marks on any algebra test.

A consolidating marathon for Section 3: linear equations, literal equations, inequalities, compound inequalities, absolute value equations and inequalities, and word problems. Work this lesson immediately before any test covering linear methods.

The two-dimensional coordinate plane: x-axis, y-axis, four quadrants, ordered pairs (x, y), and the conventions that make everyone read graphs the same way. We plot points, identify quadrants, and lay the visual foundation that every later graphing lesson, slope, parabolas, exponentials, conic sections, builds on top of.

The distance formula d equals root of (x2 minus x1) squared plus (y2 minus y1) squared and the midpoint formula M equals ((x1 plus x2) over 2, (y1 plus y2) over 2). We derive each from the Pythagorean theorem, then apply them to real coordinate problems: finding the length of a segment, the center of a circle, the midpoint of a chord. These formulas appear constantly in geometry and on the SAT.

Slope is the single most important number in linear algebra: rise over run, the steepness and direction of any line. We derive the slope formula m equals (y2 minus y1) over (x2 minus x1), work through positive, negative, zero, and undefined slopes with worked examples, and connect slope to real-world rates: wheelchair ramps, road grades, and rate-of-change in any time series.

y equals m x plus b, the most-used equation in algebra. The slope m and y-intercept b together describe any non-vertical line, and we work through reading them off the equation, plotting the line by hand, and writing the equation given a graph. The slope-intercept form is the lens through which most real-world linear models are read.

y minus y1 equals m times (x minus x1), when you know one point and the slope but not the y-intercept. We derive the point-slope form, show when it's the cleanest tool (tangent lines, parallel lines, lines through a known point), and convert between point-slope and slope-intercept forms fluently.

A x plus B y equals C, the format that makes finding x- and y-intercepts trivial, and the form most often used in linear programming and systems of equations. We cover converting to and from standard form, identifying intercepts at a glance, and choosing the right form (slope-intercept, point-slope, or standard) for the situation at hand.

Two lines are parallel if their slopes are equal, perpendicular if their slopes are negative reciprocals (m1 times m2 equals negative 1). We prove both rules, work through examples writing the equation of a parallel or perpendicular line through a given point, and connect these rules to geometric proofs and analytic geometry questions on standardized tests.

Graphing y greater than 2x plus 1 or x minus 3y less than or equal to 6, the visual extension of linear inequalities to two dimensions. We cover boundary lines (solid vs dashed), choosing the correct half-plane via test point, and the special cases (horizontal and vertical inequalities). These graphs are the visual foundation for systems of inequalities and linear programming in Section 6.

Cell phone bills, taxi fares, hourly wages, depreciation schedules, the world is full of relationships that are linear over a meaningful range. We model each from a verbal description, write the equation, graph it, and answer real questions: When does this plan become cheaper? What's the cost at 100 minutes? When will the car be worth $5,000? The modeling skill transfers directly to physics, economics, and data science.

A consolidating run through Section 4: coordinate plane basics, distance and midpoint, slope, all four line forms, parallel and perpendicular, graphing inequalities, and real-world modeling. Work this lesson before any test covering linear graphing or as a refresher before Section 5 (functions).

The definition of a function: a rule that assigns exactly one output to each input. We cover the vertical line test, the difference between functions and relations, and the crucial idea that "function" is just a name for any predictable input-output relationship. Functions become the central object of every later section, so we build the intuition slowly with multiple representations: tables, graphs, equations, and verbal descriptions.

f(x), g(x), h(t), the notation that lets us name and evaluate functions cleanly. We cover what f(3) means, how to evaluate at a specific input, the difference between f(x plus 1) and f(x) plus 1, and the standard algebra for working with named functions. Every later lesson, composition, inverses, transformations, depends on fluency with this notation.

Domain is "all valid inputs", range is "all possible outputs". We find both for linear, quadratic, square root, rational, and absolute value functions, working through the rules (no division by zero, no square roots of negatives in the real numbers, etc). Domain and range questions appear on every algebra test and are the foundation of asymptote analysis later in the course.

Functions of the form f(x) equals m x plus b. We connect everything from Section 4 (slope, intercepts, line forms) to function notation, see why linear functions have constant rate of change, and use them as the simplest playground for testing every later technique (composition, inverses, transformations).

The composition (f after g)(x) equals f(g(x)), applying one function to the output of another. We work through composition with linear, quadratic, and other functions, build the standard "inside-out" computation routine, and see why composition is the gateway to the inverse function technique that follows.

The inverse f-inverse "undoes" what f does. We cover the definition, the algebraic technique to find an inverse (swap x and y, solve for y), the graphical reflection across the line y equals x, and the horizontal-line test that determines when an inverse exists. Inverse functions are essential for solving exponential and logarithmic equations later.

Functions defined by different rules on different intervals: taxi fare, tax brackets, postage rates, the absolute value function itself. We cover writing piecewise definitions, evaluating at specific inputs, and graphing piecewise and step functions including the famous greatest-integer (floor) function.

f(x) equals the absolute value of x as the canonical V-shaped graph. We cover its key features (vertex, symmetry, slope on each side), transformations (shifting, stretching, reflecting), and connect it to absolute value equations and inequalities from Section 3. The absolute value graph is the best small example for introducing function transformations cleanly.

Shift, stretch, reflect, the three operations that turn any parent function into a family of related curves. We cover horizontal and vertical shifts, vertical and horizontal stretches and compressions, and reflections across the x and y axes, working through the rules in y equals a times f(b times (x minus h)) plus k. Transformations show up on every standardized test and are the visual key to graphing exponentials, logs, and trig in pre-calculus.

A function is even if f(negative x) equals f(x) (symmetric about the y-axis), odd if f(negative x) equals negative f(x) (symmetric about the origin). We test functions algebraically and graphically, see why most functions are neither, and apply even and odd analysis to simplify problems in calculus and physics later.

Adding, subtracting, multiplying, and dividing functions, the algebraic operations on functions themselves. We cover (f plus g)(x), (f times g)(x), (f over g)(x), and the new domain restrictions that result. This lesson closes the function toolkit before we apply it to systems and the rest of the course.

A consolidating marathon for Section 5: function definition, notation, domain and range, composition, inverses, piecewise, absolute value, transformations, even and odd, and combining functions. Work before any test covering functions or as a comprehensive review before pre-calculus.

A system of equations is two or more equations sharing the same variables, and we want the (x, y) that satisfies all of them at once. We introduce the three solution types: exactly one solution (intersecting lines), no solution (parallel lines), infinite solutions (same line). And the visual interpretation as the intersection of graphs. The whole section builds from this picture.

The substitution method for systems: solve one equation for one variable, substitute into the other. We work through systems of all three solution types, identify when substitution is the cleanest method (when one equation already has an isolated variable), and walk through the standard "show every step" format that prevents algebraic slips.

The elimination method (also called addition or linear combination): add or subtract equations to cancel a variable. We cover when elimination wins over substitution (when both equations are in standard form), how to multiply equations to set up cancellation, and the algebraic shortcuts that real engineers and scientists use to solve linear systems quickly.

A diagnostic for choosing between graphing, substitution, and elimination for any 2-variable system. We work through 6 to 8 systems of varying types and apply the diagnostic in real time. The right-method choice cuts solution time by a factor of two on tests, where speed matters as much as accuracy.

Recognizing the two non-standard outcomes algebraically: no solution gives a contradiction (5 equals 8), infinite solutions give an identity (8 equals 8). We work through examples of each, connect them to the geometric pictures (parallel lines, identical lines), and warn about the most common student mistake: stopping at the contradiction without writing "no solution" as the conclusion.

Three equations in three unknowns, the next level up. We cover the systematic approach (eliminate one variable to reduce to a 2-variable system, then back-substitute), the geometric interpretation as planes in 3D space, and the same three solution types extended (one point, no point, an entire line of points). Three-variable systems are the foundation of linear algebra and appear in any STEM major.

Word problems where systems are the natural tool: mixture problems, money and coin problems, distance and rate problems with multiple participants, age problems, supply and demand. We cover the standard "let x equal ... and let y equal ..." setup, write both equations from the verbal information, then solve. Systems word problems are tested heavily on the SAT, ACT, and GRE.

Systems of inequalities, finding the (x, y) region that satisfies all conditions at once. We cover graphing each inequality as a half-plane, finding the overlap (the feasible region), and identifying its corners (vertices). This lesson is the geometric foundation for linear programming in the next lesson and for constrained optimization in any field.

The classic optimization technique: maximize or minimize a linear objective function subject to linear constraints. We cover setting up the constraints, graphing the feasible region, evaluating the objective at each corner, and identifying the optimum. Linear programming is the workhorse of operations research, supply chain, finance, and modern AI, and it's just systems of inequalities with one extra step.

A consolidating marathon for Section 6: systems of equations (substitution, elimination), three-variable systems, systems of inequalities, and linear programming. Work before any test covering systems or as a review before tackling polynomials.

A polynomial is an expression of the form a sub n times x to the n plus ... plus a sub 1 times x plus a sub 0, the building block of everything from quadratics to high-degree functions in physics and engineering. We cover degree, leading coefficient, standard form, like terms, and the vocabulary you will use for the rest of the section. A solid foundation here pays off through every later polynomial method.

Adding and subtracting polynomials by combining like terms, the simplest polynomial operation, but with traps in sign handling and lining up degrees. We work through examples in single and multiple variables, including the famous distribute-the-negative trap on subtraction.

The distributive property scaled up, multiplying any polynomial by any other polynomial. We cover monomial times polynomial, binomial times binomial (FOIL), trinomial times trinomial, and the systematic approach for any size pair. This lesson is the foundation for the special products and factoring in Sections 7 and 8.

The five identities every algebra student needs in muscle memory: (a plus b) squared equals a squared plus 2 a b plus b squared, (a minus b) squared equals a squared minus 2 a b plus b squared, (a plus b)(a minus b) equals a squared minus b squared, (a plus b) cubed, and (a minus b) cubed. We derive each, work through examples, and show how recognizing these patterns saves significant time on factoring later.

Long division of polynomials, the same algorithm as numeric long division, just with x's instead of digits. We work through divisions step by step, identify the quotient and remainder, and connect the result to the division algorithm: dividend equals divisor times quotient plus remainder. This lesson sets up synthetic division and the remainder theorem.

A faster shortcut to long division when dividing by (x minus c). We cover the synthetic-division tableau, read the quotient and remainder directly from the bottom row, and apply synthetic division to factor candidates from the rational root theorem in later lessons. Synthetic division cuts polynomial-division time by 60 percent or more once you're fluent with it.

The remainder theorem: P(c) equals the remainder when P(x) is divided by (x minus c). The factor theorem: (x minus c) is a factor of P(x) if and only if P(c) equals 0. We prove both theorems, apply them to test factor candidates, and use them as the foundation for the rational root theorem in Lesson 9.

Graphing polynomial functions by combining four key features: end behavior (from the leading term), y-intercept (the constant term), x-intercepts (the real roots), and behavior at each root (cross vs bounce, by multiplicity). We work through examples up to degree 5 and develop the standard sketch-from-features routine.

The rational root theorem: every rational root of a polynomial with integer coefficients has the form p over q where p divides the constant term and q divides the leading coefficient. We list candidates, test each with synthetic division, and reduce the polynomial as roots are found. This is the systematic algorithm for finding rational roots of any polynomial.

Combining everything from Section 7 into a master algorithm: find all rational roots first (rational root theorem plus synthetic division), reduce the polynomial, finish the leftover quadratic with the quadratic formula. The fundamental theorem of algebra guarantees exactly degree-many roots in the complex numbers, and complex roots of real polynomials always come in conjugate pairs. By the end, no polynomial can hide a root from you.

A consolidating marathon for Section 7: polynomial operations, special products, long and synthetic division, the remainder and factor theorems, graphing, rational roots, and finding all roots. Work before any test covering polynomials or as a comprehensive review before factoring (Section 8).

The first move in any factoring problem: pull out the greatest common factor (GCF). We cover identifying the GCF for monomials and polynomials, factoring it out cleanly, and the most common student mistakes (forgetting the GCF, taking too small a GCF). Every later factoring method assumes you've already done this step.

The four-term factoring method: group terms in pairs, factor each pair, then factor out the common binomial. We work through examples that come from the expansion of binomial times trinomial products, and show how grouping is the foundation for factoring trinomials with leading coefficient greater than 1.

Factoring trinomials of the form x squared plus b x plus c. Find two numbers that multiply to c and add to b. We work through positive and negative coefficients, the prime case (when no such factoring exists), and the standard "guess and check" technique organized as a fast lookup.

Trinomials of the form a x squared plus b x plus c where a is not equal to 1, the harder case. We cover the AC method (multiply a and c, factor by grouping), the alternative trial-and-error approach, and the systematic way to choose between them. By the end, no trinomial, no matter the leading coefficient, will block you.

The pattern a squared minus b squared equals (a minus b)(a plus b), one of the most-tested factoring patterns on every algebra exam. We cover recognizing the pattern (both terms perfect squares, separated by minus), repeated applications (factor again if a perfect-square difference appears in a factor), and the famous "sum of squares is NOT factorable over the reals" exception.

The two cubic factoring patterns: a cubed plus b cubed equals (a plus b)(a squared minus a b plus b squared) and a cubed minus b cubed equals (a minus b)(a squared plus a b plus b squared). We derive each, recognize them by the perfect-cube terms, and apply them to expressions that don't look cubic at first (with the right substitution). Cubic-pattern recognition is the most-skipped tool in factoring.

The pattern a squared plus or minus 2 a b plus b squared equals (a plus or minus b) squared. Recognizing perfect square trinomials by the doubled middle term. We cover recognition, factoring, and the connection to the special-products identities from Section 7. Perfect-square recognition is the foundation for completing the square in Section 11.

Real factoring problems mix multiple methods. We cover the standard decision tree: GCF first, then count terms (2 equals difference of squares or cubes; 3 equals trinomial; 4 equals grouping), then check each factor for further factoring. By the end, you will handle any factoring problem that appears on a test or in higher math.

The zero product property: if A times B equals 0, then A equals 0 or B equals 0. We use it to solve quadratic and higher-degree polynomial equations by factoring, working through examples that connect Section 7 polynomial methods, Section 8 factoring methods, and the equation-solving toolkit you've built since Section 3.

A consolidating marathon for Section 8: GCF, grouping, trinomial factoring (both leading-coefficient cases), difference of squares, sum and difference of cubes, perfect square trinomials, combining methods, and solving by factoring. Work before any test covering factoring or as a review before rational expressions.

A rational expression is a polynomial divided by a polynomial, a fraction with x's. We cover factoring numerator and denominator, canceling common factors, and the crucial domain restrictions (excluded values where the original denominator was zero). The factoring toolkit from Section 8 is essential here.

Multiplication and division of rational expressions, same rules as numeric fractions, with factoring for cancellation. We cover both operations with single and multiple-factor numerators and denominators, the cross-cancellation shortcuts, and the standard "factor everything first" routine that prevents simplification errors.

The hardest rational-expression operation: adding and subtracting requires a common denominator, and finding the LCM of polynomial denominators is more involved than with numbers. We cover the systematic LCM technique, equivalent-fraction conversion, and the simplification of the result. This skill transfers directly to partial fraction decomposition in calculus.

Fractions inside fractions, complex fractions look intimidating but reduce with two standard techniques: the LCD method (multiply numerator and denominator by the overall LCD) and the divide-out method (rewrite as one fraction divided by another). We cover both and show when each is faster.

Equations containing rational expressions. We multiply both sides by the LCD to clear denominators, solve the resulting polynomial equation, and check for extraneous solutions (apparent solutions that make a denominator zero). The extraneous-solution check is THE most-skipped step in rational equation solving and it's the difference between right and wrong answers.

Direct variation (y equals k x), inverse variation (y equals k over x), and joint variation (y equals k x z). We set up each from a verbal description, find k from given data, and use the formula to predict new values. Variation problems are foundational in physics (Hooke's law, Coulomb's law, gravity) and finance (loan interest, depreciation).

Real applications of rational expressions: work-rate problems (two pumps filling a tank), distance-rate-time with averaged speeds, optics lens equations, mixing problems. Each model uses the same algebraic techniques you've just built, but the verbal-to-equation translation is the crucial bridge.

Graphing rational functions by identifying their key features: vertical asymptotes (denominator zeros), horizontal or oblique asymptotes (degree comparison), x-intercepts (numerator zeros), holes (canceled common factors). We cover the systematic graphing routine and apply it to the most-tested function shapes.

Decomposing a single rational expression into a sum of simpler ones, the inverse of common-denominator addition. Partial fractions are the foundation for integral calculus's hardest technique, and even at the algebra level they have applications in signals processing and control systems engineering.

A consolidating marathon for Section 9: simplifying, multiplying and dividing, adding and subtracting, complex fractions, rational equations, variation, applications, graphing rational functions, and partial fractions. Work before any test covering rationals or as a review before radicals.

The square root, cube root, and nth root, the inverse operations to powers. We cover the radical symbol, the radicand, the index, and the algebra rules for combining radicals. Radicals are both essential algebra notation and a frequent source of student confusion, so we build them carefully.

Simplifying root 48 to 4 root 3, pulling out perfect-square factors. We cover the systematic technique for square roots and higher-index radicals, the "extracting perfect powers" routine, and the simplest form conventions tested on every algebra exam.

You can only combine radicals with the same radicand and index, like terms. We cover identifying like radicals, simplifying first to find them, and the most common student mistake (treating root a plus root b as root of (a plus b), which it never equals).

Multiplying and dividing radicals using the product and quotient rules. We cover combining radicals under one root, splitting after multiplication, and the FOIL pattern for binomial radical expressions. The techniques set up rationalizing denominators in the next lesson.

The convention: no radicals in the denominator of a final answer. We cover rationalizing single-term denominators (multiply by the radical) and binomial denominators (multiply by the conjugate), and apply them to standardize answers in the form expected by tests and textbooks.

The connection between radicals and exponents: a to the (1 over n) equals nth root of a, and a to the (m over n) equals (nth root of a) to the m. We cover the conversion rules, applying exponent laws to rational-exponent expressions, and the simplification techniques that work uniformly across all radical operations.

Equations containing radicals require isolating the radical, then squaring (or raising to the appropriate power) both sides. We cover the standard technique, the crucial extraneous-solution check (squaring can introduce false solutions), and the multi-radical equations that appear on harder tests.

Cube roots and nth roots beyond the square root: their domain (cube roots accept negative inputs; even-index roots don't), their graphs, their algebraic rules. We cover simplification, equations, and the way higher-index radicals show up in finance (rule of 72, return of return) and physics (mean lifetimes).

Real applications of radicals: the Pythagorean theorem and distance formula, free-fall time from height, period of a pendulum, signal amplitude, standard deviation. Each connects an algebraic radical to a physical or financial quantity.

A consolidating marathon for Section 10: simplifying, four operations on radicals, rationalizing, rational exponents, radical equations, higher-index roots, and applications. Work before any test covering radicals or as a review before quadratics.