Suppose two modeled contributions are 1/(x−1) and 1/(x+1). Combining them does not mean adding their denominators. At x=2 the contributions are 1 and 1/3, so their sum must be 4/3. The tempting expression 2/(2x) would give 1/2 instead. Use this numerical failure to motivate a common denominator rather than memorizing an unexplained rule.
Predict
Predict the excluded inputs before simplifying anything. Then predict whether cancellation can make either original contribution meaningful at an excluded input. A shorter formula may have a larger natural domain, but that does not alter what the original operation meant. For division, make one further prediction: besides requiring the divisor to be defined, what value must it avoid?
Experiment
Track arithmetic and domains in parallel. Factor denominators before canceling and carry exclusions through every row.
Operation
Rewritten formula
Original exclusions
1/(x−1)+1/(x+1)
2x/[(x−1)(x+1)]
x=1 or x=−1
1/(x−1)−1/(x+1)
2/[(x−1)(x+1)]
x=1 or x=−1
[(x²−1)/(x−2)]·[1/(x+1)]
(x−1)/(x−2)
x=2 or x=−1
Verify each row at x=3. In the product row, the simplified formula alone has a value at x=−1, but the original second factor does not. Mark this as an extension of the simplified formula rather than restoration of the original expression.
Understand
For polynomial numerators p,r and denominators q,s, the sum is (ps+rq)/(qs), the difference is (ps−rq)/(qs), and the product is pr/(qs). At an evaluated input, both q and s must be nonzero. These formulas follow by multiplying a fraction by a nonzero factor divided by itself and then applying the ordinary fraction rules. Polynomial closure ensures the results still have polynomial numerators and denominators.
Division requires more care. The quotient (p/q) divided by (r/s) becomes ps/(qr), but the original operation requires q and s nonzero and also r nonzero at that input. The shortened denominator qr does not record the original exclusion s=0. For example, [(x²−1)/(x−2)] divided by [(x+1)/(x−2)] simplifies to x−1, yet x=2 remains excluded because the operands were undefined, and x=−1 remains excluded because the divisor was zero.
Two meanings of “nonzero” matter. A denominator polynomial must not be identically zero to define a rational expression at all. Even a nonzero polynomial may vanish at specific inputs; x−2 is not the zero polynomial, but its value at 2 is zero. Likewise, a nonzero rational function can have zeros, and pointwise division excludes them. Algebraic fraction-field language identifies equivalent rational functions, but a contextual expression may still carry holes inherited from its original operations.
Cancel common factors, not terms separated by addition. The numerator x+1 and denominator x+2 have no common factor x to cancel. Testing a small permitted input can expose that error, but factoring and the distributive law explain why the cancellation rule works.
Apply
For each task, write the original restriction list first, then calculate, then compare with the simplified formula’s natural domain. When dividing by a quotient, separately check that it is defined and that it is not zero. Use a permitted numerical input as a final sanity check, not as the proof that two symbolic formulas agree everywhere on their common domain.
Continue
These arithmetic rules support rational equations, but an expression simplification is not yet an equation solution. Next, parameterized inequalities add another reason to avoid unconditional division: even a nonzero divisor can change the direction of an inequality when its sign changes.
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Evaluate 1/(x−1)+2/(x+1) at x=3.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Evaluate [(x²−4)/(x−1)] ÷ [(x+2)/(x−1)] at x=4.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Simplify 2/(x−2)+3/(x+2) and 2/(x−2)−3/(x+2), recording original domains.
Structure your argument and preserve revisions
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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.
Simplify [(x²−9)/(x²−x)]·[x/(x+3)] and state the complete original domain. Compare it with the simplified formula domain.
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.
Simplify [(x²−4)/(x−1)] ÷ [(x+2)/(x−1)]. State all restrictions before taking a reciprocal. Why is x=−2 excluded although the final formula is x−2?
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.
For polynomial expressions p/q and r/s, give formulas for their sum, difference, product and quotient. State evaluation restrictions and explain why the results are rational expressions when the divisor is not the zero rational expression.
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.
