Use distribution and structure without confusing a few matching values with an identity.
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Two fictional packing plans are priced by the number n of packages. Plan A is 4(n + 3) dollars. Plan B is 4n + 12 dollars. Calculate both for n = 2 and n = 5. Are these merely two coincidences, or do the rules agree for every allowed n?
Predict
Now compare 4(n + 3) with 4n + 3. Predict the difference before evaluating. The outside factor multiplies the extra three as well as n; leaving the three unchanged loses 9 dollars.
Experiment
Use a rectangle divided into two adjacent parts as a paper diagram. Give both parts height 4, and widths n and 3. Label the whole area 4(n + 3), and the two areas 4n and 12. This picture explains distribution for nonnegative lengths. The algebraic distributive law also applies to negative real inputs, where physical lengths no longer provide the explanation.
n
4(n + 3)
4n + 12
4n + 3
2
20
20
11
5
32
32
23
Understand
The distributive law a(b + c) = ab + ac is an identity: it holds for every real a, b and c. Subtraction is addition of a negative quantity. Thus −3(2x − 5) = −6x + 15, not −6x − 15.
Like terms have the same variable part: 6x − 2x = (6 − 2)x = 4x. But x² and x are different variable parts and cannot be combined into 2x². A useful check is x = 2: x² + x is 6, while 2x² is 8.
One counterexample disproves an identity. Several successful substitutions do not prove one. For example, x² = x works at 0 and 1 but fails at 2. A structural derivation explains why a proposed identity holds throughout its domain.
Domain matters. Canceling a factor divides by it, so it must be nonzero. The formulas (x² − 1)/(x − 1) and x + 1 agree when x is not 1. Only the second is defined at 1. Their formulas agree on the original domain, but they are not identical functions with identical domains.
Apply
Practice both simplification and error diagnosis below. In a written explanation, show the intermediate distribution or factorization rather than saying “it simplifies.” Record restrictions before cancellation.
Continue
Equivalent expressions can replace each other inside an equation without changing its meaning on the stated domain. The next lesson adds operations on both sides and asks which steps preserve the entire solution set.
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Practice and reasoning
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After simplifying 3(2x − 5) − 2(x + 4), what is the coefficient of x?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
What is the constant term in −2(3 − x) + 5x − 1?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which identity holds for every real x?
To test the claim x² = x for every real x, use x = 2. What is x² − x?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Explain precisely why (x² − 9)/(x − 3) can be replaced by x + 3 in a calculation only with a restriction. Does simplifying restore the missing input?
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.
