The decision and its conditions
A fraction whose numerator or denominator contains fractions still follows the ordinary division rule. Simplify the complete denominator before taking its reciprocal. Combining rational expressions requires a shared denominator; subtracting changes the signs of all numerator terms being subtracted. Cancellation removes a common factor only where the original expression was defined. A removable hole has a missing input and an associated output in the simplified formula. That output is useful for locating the hole, but it is not the value of the original undefined expression.
Worked example
Evaluate (1/3) ÷ (1/6 + 1/9) exactly. The denominator sum is 5/18. Divide by that sum by multiplying by 18/5. The common factor 3 cancels and the result is 6/5.
A contrasting decision
At x=4, evaluate 1/(x−1) − 1/(x+1). A common denominator is (x−1)(x+1). The numerator is (x+1)−(x−1)=2. At x=4, the value is 2/(15)=2/15.
Diagnose a competing method
The first example’s tempting answer 5/2 follows this mistake: Add fraction numerators and denominators separately in the divisor. The entire sum is the divisor; combine it before taking a reciprocal.
Transfer before revealing
The graph of y=[(x−4)(x+7)]/(x−4) has a removable hole. What is the hole’s y-coordinate in the simplified line? Pause and identify the original restrictions. Solution: The original expression excludes x=4. For allowed inputs the formula simplifies to y=x+7. The missing point has height 4+7=11; the original function is still undefined there.
Check the boundary of the method
A removable hole has a missing input and an associated output in the simplified formula. Subtract the entire second numerator after forming a common denominator. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
Evaluate (1/3) ÷ (1/6 + 1/9) exactly.
- The denominator sum is 5/18.
- Divide by that sum by multiplying by 18/5.
- The common factor 3 cancels and the result is 6/5.
Compare the other answer choices
Add fraction numerators and denominators separately in the divisor.
Divide by only the first term, changing the grouping.
Reverse the final quotient.
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.
CLEP College Algebra exam overview