Where this sits. This is one of the five algebra families — a little over half the test, and where most students have the least ground under them. Every family page on this site has the same eight parts, in the same order, so once you know the shape of one you know the shape of all of them. Where a part is empty, it says so.
DrillsSixteen drills for this family, one skill each, ten problems at a timeGo to the drills ↓An equation says two amounts are equal, and solving it means finding the number the variable stands for while keeping the balance true. One step, then two, then variables on both sides, then parentheses.
Inequalities are the same work with “at least,” “no more than,” “under” — and one rule of their own, that multiplying or dividing by a negative turns the sign around. Word problems live here too: the hard part is writing the equation, not solving it.
Quadratic equations belong here as well — the ones where the variable is squared. The test asks you to solve them, either by factoring or with the quadratic formula, and it gives you that formula on the sheet along with the standard form it fits, ax² + bx + c. Quiz 40, below, teaches both ways.
What the test asks in this family
- What an equation says, and what solving one meansQuiz 33 · The Thing Called x
- One-step equationsQuiz 33 · The Balance · The Share · Algebra Practice 1
- Two-step equationsQuiz 33 · The Balance · Algebra Practice 1 · Lines on the Grid
- Checking whether a number is the solutionQuiz 33 · Algebra Practice 1
- Writing an equation from wordsQuiz 33 · Algebra Practice 1 · Lines on the Grid
- Turning a formula around to find a different partCircles and Equations
- Inequalities, and the one rule that turns the signThe Share · Lines on the Grid
- Variables on both sidesQuiz 40
- Parentheses inside an equationQuiz 40
- Quadratic equations, solved by factoringQuiz 40
- The quadratic formulaQuiz 40
All 11 have something on this site.
Seven questions across this family. Type an answer and press Check. You get the reasoning for every one, right or wrong.
- Solve for x: 5x − 8 = 27 x = 7. Undo what was done to x, in reverse order. The 8 was taken away, so add it to both sides: 5x = 35. The x was multiplied by 5, so divide both sides by 5: x = 7. Whatever you do to one side you do to the other, and the balance stays true.
- Solve for x: 4x + 3 = 2x + 15 x = 6. With variables on both sides, first get them together. Take 2x from both sides: 2x + 3 = 15. Then take 3 from both: 2x = 12. Then divide by 2: x = 6. Check it by putting 6 back in — 27 on both sides.
- Solve for x: 3(x − 2) = 18 x = 8. Two ways, both right. Multiply out first: 3x − 6 = 18, so 3x = 24 and x = 8. Or divide both sides by 3 straight away: x − 2 = 6, so x = 8. The second is quicker when the number outside divides the other side cleanly.
- Solve: 5 − 2x > 11 x < −3. Solve it exactly like an equation until the last step: take 5 from both sides to get −2x > 6, then divide both sides by −2. That is the one rule of its own — dividing by a negative turns the sign around, so it becomes x < −3. Test it with −4: 5 − 2(−4) = 13, which is greater than 11.
- A plumber charges $75 to come out, plus $50 an hour. The bill was $325. How many hours? 5 hours. Write the equation before you solve anything: 75 + 50h = 325. Take the call-out off both sides: 50h = 250. Divide: h = 5. The hard part of a word problem is the writing, not the solving, and the sentence usually gives the equation in order.
- Is x = 3 a solution of 4x − 5 = 7? Yes. Put it in and see: 4(3) − 5 = 12 − 5 = 7, which is what the right-hand side says. Checking a solution is a question the test asks outright, and it is also the check you can run on any answer you are unsure of.
- Solve by factoring: x² + 5x + 6 = 0 x = −2 and x = −3. A quadratic usually has two answers. Factor the left side into (x + 2)(x + 3) = 0 — look for two numbers that multiply to 6 and add to 5. If two things multiply to zero, one of them must be zero, so x + 2 = 0 or x + 3 = 0. The quadratic formula is on the sheet they give you and would find the same two answers.
Missed one? Quiz 33: Words to Symbols in Learn it covers writing an equation from words and solving it in one or two steps. Quiz 40: Two Answers covers variables on both sides, parentheses in an equation, quadratics by factoring and the quadratic formula. The three rooms under Work it by hand build equation-solving from the beginning, and the two practice sheets under On paper have the drill. Every part of this family now has a lesson.
Want more than these seven questions? Flyover I asks about all fourteen families in one sitting and ends in a grid that says where to start; Flyover II asks them again later, with different questions, so you can see what has moved.
- Quiz 33Words to SymbolsTurning a sentence into an equation, and solving it by undoing what was done: one step, then two.
- Quiz 40Two AnswersVariables on both sides, parentheses in the way, quadratics solved by factoring, and the formula for the ones that will not factor.
Two quizzes and three rooms, and between them this family is complete. Start with Quiz 33. Quiz 40 needs factoring, so do Quiz 39 before it if you have not.
Sixteen short drills, one skill each, in order. Each starts with two worked examples, then gives you ten problems (or five) and the reasoning for every answer. Each one prints, with the answer key on the last page. When you get all ten right, or all five right twice in a row, the drill gets a check mark on the Drills page.
- 12.01One-step equations: adding and subtracting
- 12.02One-step equations: multiplying and dividing
- 12.03Two-step equations
- 12.04The variable on both sides
- 12.05Equations with parentheses
- 12.06Equations with fractions · calculator allowed
- 12.07Check a solution by putting it back in
- 12.08Solve a formula for one of its variables
- 12.09Write and solve an equation from a word problem · calculator allowed
- 12.10Inequality signs, and graphs on a number line
- 12.11One-step inequalities, including flipping the sign
- 12.12Multi-step inequalities
- 12.13Inequality word problems · calculator allowed
- 12.14Quadratics by taking the square root (x² = 49)
- 12.15Quadratics by factoring
- 12.16The quadratic formula · calculator allowed
- Algebra, one move at a time · three rooms, in order.
- 1. The Thing Called x · where algebra came from, and what a variable is standing in for.
- 2. The Balance · an equation on a balance scale that you move yourself.
- 3. The Share · sharing the blocks out, which is division with a variable in it.
- Circles and Equations · its first three rooms turn an equation around: if 2 × 3 = 6, then 2 = 6 ÷ 3 and 3 = 6 ÷ 2, and the same move works on any formula.
- Algebra Practice 1 · practice for the three rooms, solving for the unknown one step at a time, with the key at the end.
- Khan Academy, math · free videos and practice. Search it for one-step equations, two-step equations, inequalities, algebra word problems. These are the clips we start from in class.
- Math Antics · short, plain videos on the basics.
- Light & Salt Learning · free GED classes online, with playlists by topic.
- Get Sum Math · GED math on video, one topic at a time.
- Kaplan GED Test Prep and the McGraw-Hill Mathematical Reasoning Workbook · look up equations and inequalities in the contents of either.
On to lines, slope, and systems. An equation with two variables in it draws a line, and a system is two lines at once. That is the whole idea, and the equation-solving you have just done is the tool for it.