A hole is not repaired
The quotient (x²−16)/(x−4) factors to (x−4)(x+4)/(x−4). It equals x+4 for x≠4, but the original quotient has no value at four. Its graph approaches height eight there. The excluded input four, the nearby height eight and the numerator zero −4 answer different questions.
Solve and test a rational equation
For 10/(x−3)=2, first exclude x=3. Multiplying by the nonzero denominator gives 10=2(x−3), then x=8. Substitution gives 10/5=2, confirming the solution is allowed. This check verifies both the equation and the restriction; merely solving the cleared equation would not do both.
Why squaring can add a root
Consider √(2x+24)=x. The right side forces x≥0. Squaring gives x²−2x−24=0, whose roots are six and −4. Six satisfies the original equation. At −4 the square root is positive four, not negative four, so that candidate is extraneous. Squaring preserved equality in the forward direction but erased sign information.
Transfer and check
Find the real solution of √(2x+8)=x. The squared candidates are four and −2; only four works. Then list exclusions for 1/[(x−2)(x+5)]: two and −5. Explain why neither their sum nor their product is itself the definition of the domain restriction.
The expression (x²−9)/(x−3) simplifies to x+3 for allowed inputs. Which input is still excluded from the original expression?
- Factor the numerator as (x−3)(x+3).
- The original denominator is zero at x=3.
- Cancellation gives the same values only where x≠3; it cannot restore this input.
Compare the other answer choices
Assume every rational expression excludes zero.
Report the limiting height at the hole instead of its input.
Use the other zero of the numerator, whose denominator is nonzero.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
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