Preserve the solution set, classify exceptional cases and check the original equation.
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A fictional club has 29 dollars after collecting 3 dollars from each member and paying a 7-dollar fee. Let n be the number of members. Translate the situation into 3n − 7 = 29 and predict n before solving.
Predict
To undo the fee, should you subtract another 7 or add 7? Think of reconstructing the earlier amount: before paying 7, the club had 36 dollars. Dividing 36 by 3 gives 12 members.
Experiment
Write a two-column calculation with the left and right sides. At every row, record the operation applied to both.
Equation
Reason
3n − 7 = 29
Original model
3n = 36
Add 7 to both sides
n = 12
Divide both sides by nonzero 3
Check the original: 3(12) − 7 = 29. Now change the total to 30. Algebra gives n = 37/3, but a whole-member model has no feasible solution. Exact arithmetic and context must both pass.
Understand
A solution is an input making the original equality true. Adding the same expression to both sides is reversible. Multiplying or dividing by a known nonzero constant is reversible. Expanding equivalent expressions also preserves the solutions.
For 2(x + 3) = 5x − 9, expand to 2x + 6 = 5x − 9. Subtract 2x, add 9, and divide by 3: 15 = 3x, so x = 5. Checking gives 16 on each original side.
Fractions require every term to participate. For x/2 + 1 = 4, multiplying by 2 yields x + 2 = 8, not x + 1 = 8.
Not every linear-looking equation has one solution. After collecting terms, ax = b has one solution b/a when a is nonzero. If a = 0 and b = 0, every real input works. If a = 0 and b is nonzero, none works. Never attempt to classify these cases by dividing by zero.
Dividing by a variable expression requires a case split. Dividing x(x − 2) = 0 by x would lose the legitimate zero solution.
Apply
Solve the practice equations and check candidates in the originals. Use “none” for an empty finite solution set. An identity has infinitely many solutions and is handled by a choice or written response, not an empty-set entry.
Continue
An inequality also has a solution set, but order introduces another rule: multiplying by a negative quantity reverses its direction. Keep your habit of recording operations and checking representative inputs.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Solve 3(2x − 1) + 4 = 2x + 13.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Solve (x − 2)/3 + (x + 1)/2 = 4.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Classify 4(x + 2) = 4x + 8 over the real numbers.
Give the real solution set of 5x − 2 = 5x + 7. Type none if empty.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
A learner solves x(x − 2) = 0 by dividing by x and reports only x = 2. Repair the reasoning and explain why division was unsafe.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
