Algebra stops being a collection of tricks the moment you notice how little actually changes from one topic to the next. You do the same thing to both sides. You keep the statement true. You check the answer against the problem you were given, not against your own working. Everything in this chapter — 23 lessons, from a first linear equation to a rational inequality with a hole in it — is that idea applied to expressions that get progressively less friendly.
The lessons below are in reading order. Each one assumes the ones above it and nothing below it, so if a step ever feels like it appeared from nowhere, the explanation is almost always one lesson earlier.
Start Here: What an Answer Actually Is
Solutions and Solution Sets — worth ten minutes even if it looks obvious. It pins down what it means for a value to satisfy an equation, how to verify one by substitution, and how to write a solution set properly. Everything after this assumes you can say whether a number is or is not a solution, and the habit of checking by substitution is the single thing that catches the most mistakes later on.
Linear Equations
The whole balancing method, established on the simplest possible case before anything harder arrives.
- Linear Equations — distributing, combining like terms, clearing fractions and getting the variable alone. Also the two odd outcomes that confuse people the first time: an identity that is true for every value, and a contradiction that is true for none. · Practice
- Applications of Linear Equations — the five-step process for turning a sentence into an equation, applied to numbers, geometry and percentages.
- Equations with More Than One Variable — solving a formula for one letter while treating the rest as constants. Nothing new algebraically; it just feels new because the answer contains letters.
- Linear Equations with Fractions — multiplying through by the LCD to clear every denominator at once, rather than fighting the fractions term by term. · Practice
Word Problems
Same algebra, harder translation. The difficulty here is almost never the solving.
- Word Problems — choosing what
stands for, writing every other unknown in terms of that same , and checking the answer against the words rather than the equation. · Practice - Distance, Rate, and Time Problems —
with a row-per-object chart. Covers objects meeting, one catching another, round trips, and currents and headwinds. · Practice - Work Problems — rates add. One worker who finishes in
hours contributes of the job each hour, and the parts sum to one whole job. · Practice - Mixture Problems — balancing the pure amount of one ingredient before and after mixing, which also covers coin and ticket problems. · Practice
- Solving for a Variable — rearranging a formula properly, including the awkward cases where the target appears twice and has to be factored out, or sits under a root. · Practice
Quadratic Equations
The second power arrives, and with it four different solving methods and the judgement to pick one.
- Quadratic Equations — standard form, the zero-product property, factoring and the square-root property, plus a guide to which method suits which equation. · Practice
- Completing the Square — the method that always works, why the number you add is
, and how it produces vertex form. · Practice - The Quadratic Formula — the universal fallback, reading
, and without sign errors, and using the discriminant to predict the answers before solving. · Practice - Applications of Quadratic Equations — word problems that end in a quadratic: areas, uniform borders, projectile height and revenue models, including how to reject a root the context makes impossible. · Practice
- Equations Reducible to Quadratic Form — spotting a quadratic hiding inside a quartic or a fractional-exponent equation, and substituting
to expose it. · Practice - Equations with Radicals — isolating the root and raising both sides to a power, plus the extraneous-solution check that is genuinely mandatory here. · Practice
Inequalities
Everything above, repeated with one extra rule and a different way of writing the answer.
- Linear Inequalities — the same steps as a linear equation, plus the rule that multiplying or dividing by a negative flips the direction of the sign. · Practice
- Compound Inequalities — statements joined by “and” or “or”, the three-part form
, and the difference between an overlap and a union. · Practice - Interval Notation — brackets against parentheses, unbounded intervals, unions, and converting between inequalities, graphs and intervals. This is the format the rest of the chapter answers in. · Practice
- Polynomial Inequalities — the sign-chart method: move everything to one side, factor, find the critical values and test one point per interval. · Practice
- Rational Inequalities — the same chart with one addition that catches people out: a value making the denominator zero is a critical value too, and it is always excluded from the answer. · Practice
Absolute Value
- Absolute Value Equations — isolating the bars and splitting into two cases, plus what happens when the other side is zero or negative. · Practice
- Absolute Value Inequalities — the same distance idea turned into a compound inequality: “less than” gives one interval, “greater than” gives two pieces. · Practice
How Each Lesson Is Built
They all follow the same shape, so you can navigate one you have not read before. The rule is stated plainly first, then derived rather than asserted, then applied in several fully worked examples. After that comes a section on the mistakes that actually cost marks on that topic, a quick-reference table you can skim before a test, and a set of practice problems whose answers are hidden until you open them.
Where a lesson has a matching set of practice problems, the link sits at the top of the page. Those are worth using once the method makes sense — reading an explanation and reproducing it are different skills, and only the second one survives an exam.
If You Are Not Sure Where to Start
Start at Solutions and Solution Sets and read down. If linear equations already feel comfortable, jump to whichever section names the thing giving you trouble — the sections above are independent enough to enter at the top of any one of them, as long as you have the linear material behind you.
If a step in an early lesson refers to factoring, exponent rules or radicals as though you already know them, that groundwork lives in Algebra Preliminaries, the chapter before this one. The full list of published lessons is on the Algebra page.