Goal and prerequisites
Use common denominators and factor cancellation without losing excluded inputs. Review algebra methods before starting.
Method and its conditions
Factor numerator and denominator before cancellation. Cancel whole nonzero factors, not terms within sums. When adding fractions, construct a common denominator and multiply each numerator by its missing factor. List exclusions before solving an equation.
Worked example: show the decisions
3/(x+1)+5/(x−1) has denominator (x+1)(x−1). Its numerator is 3(x−1)+5(x+1)=8 x+2. Inputs −1 and 1 stay excluded. Separately, (x²−16)/(x+4)=x−4 for x≠−4; the reduced expression does not restore the excluded input.
Contrast: diagnose an incorrect approach
Canceling x in (x+3)/(x+1) is invalid because x is part of a sum, not a common factor. When solving 4/x+1/2=3/2, subtract the half first: 4/x=1, so x=4. Multiplying only one term by the common denominator changes the equation.
Transfer: try a changed situation
Combine 2/(x+1)+1/(x−1) into one fraction and name its excluded inputs.
Transfer solution and interpretation
The numerator is 2(x−1)+(x+1)=3 x−1 and the denominator is x²−1. Exclude x=−1 and x=1. Substitution at x=2 gives 5/3 in both representations.
Keep hidden and visible restrictions together
For ((x−4)(x+4))/((x−4)(x−6)), the original denominator excludes both 4 and 6. The reduced expression (x+4)/(x−6) still excludes 6, and cancellation does not restore 4. Thus the sum of the two excluded inputs is 10; neither checking only the original canceled factor nor checking only the reduced denominator is sufficient.
Combining 3/(x+1)+5/(x−1) over denominator (x+1)(x−1) gives numerator cx+d. Find c.
- Use the common denominator (x+1)(x−1), retaining x≠−1 and x≠1.
- The numerator is 3(x−1)+5(x+1)=3 x−3+5 x+5=8 x+2.
- The requested coefficient of x is 8.
Compare the other answer choices
Report the constant coefficient instead of the x coefficient.
Keep only the first fraction’s x coefficient.
Multiply the numerator coefficients instead of adding cross-products.
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.
0580 syllabus for exams in 2025–2027