I’m revising algebraic fractions to strengthen my fundamentals, and I keep tripping over when it’s valid to cancel things.
Here’s my completely wrong attempt (so you can see my confusion):
– I tried to simplify (x+6)/(x+2) by “cancelling the x” and got 6/2 = 3.
– Quick number check: if x = 4, the original is (4+6)/(4+2) = 10/6, but my result is 3. So I know this move is illegal – I just can’t articulate exactly why.
What I’m trying to pin down is a reliable step-by-step rule:
– When is it actually okay to cancel something that appears in the numerator and denominator? Does it have to be a common factor of the entire numerator and the entire denominator (not just part of a sum)?
– Should I always factor first before even thinking about cancelling? For example, with (x^2 − 4)/(x − 2) or (x+2)/(2x+4), what’s the correct thought process to decide if anything can be cancelled?
– I also mess up when adding fractions. I once did this (wrong): x/(x+2) + 1/(x+3) → (x+1)/((x+2)+(x+3)) = (x+1)/(2x+5). Plugging in x = 1 gives 1/3 + 1/4 vs 2/7, which don’t match. What’s the clear, systematic way to handle addition so I don’t fall into that trap?
– Do I need to keep track of any excluded values before or after cancelling, or is that a separate step?
If anyone can outline a careful checklist (factor? identify common factors? only then cancel? track restrictions?) and explain why my “cancel the x” idea fails in (x+6)/(x+2), that would really help me build the right habits.
Ready to make maths more enjoyable, accessible, and fun? Join a friendly community where you can explore puzzles, ask questions, track your progress, and learn at your own pace.
By becoming a member, you unlock:
3 Responses
I’m tempted to say you can “cancel the x” in (x+6)/(x+2) since x shows up in both places, so maybe it should drop out and leave 6/2 = 3. But that only works when the thing you cancel is a common factor of the whole numerator and the whole denominator. Here x is part of a sum, not a factor multiplying the entire expression. The safe habit is: factor first; then you may cancel only factors that multiply the entire numerator and entire denominator. Plus signs are a stop sign for cancellation. Also note domain restrictions come from the original denominator and must be kept even after cancelling.
A quick worked example: (x^2−4)/(x−2) factors to (x−2)(x+2)/(x−2). Now (x−2) is a common factor, so it cancels and you get x+2, with the restriction x≠2 from the original denominator. Another one: (x+2)/(2x+4) = (x+2)/(2(x+2)) → 1/2 after cancelling (x+2), with x≠−2. By contrast, in (x+6)/(x+2) there is no common factor after factoring (nothing nontrivial to factor out), so nothing cancels.
For addition, don’t add denominators. Find a common denominator, combine the numerators, then factor and cancel if possible, keeping all original restrictions. Example: x/(x+2) + 1/(x+3) = [x(x+3)+1(x+2)]/[(x+2)(x+3)] = (x^2+4x+2)/((x+2)(x+3)). The numerator doesn’t share a factor with the denominator, so you stop here, with x≠−2,−3.
Rule of thumb I wish I’d learned sooner: you can only cancel common factors that multiply the entire numerator and denominator-never across addition/subtraction (the plus sign is a wall), so factor first, then cancel, and record excluded values from any denominator/cancelled factor (e.g., (x+6)/(x+2): nothing cancels; (x^2−4)/(x−2) = (x−2)(x+2)/(x−2) = x+2 with x≠2; (x+2)/(2x+4) = (x+2)/[2(x+2)] = 1/2 with x≠−2). For addition, build a common denominator and combine numerators: x/(x+2)+1/(x+3) = [x(x+3)+(x+2)]/[(x+2)(x+3)] = (x^2+4x+2)/[(x+2)(x+3)]-I once “cancelled the x’s” and my plug-in check ratted me out, and this helped me fix the habit: https://www.khanacademy.org/math/algebra/rational-expressions-functions.
Quick compass: you can only cancel common factors that multiply the whole numerator and denominator-never pieces inside a sum-so the x’s in (x+6)/(x+2) are “glued by +” and can’t cancel; habit: note restrictions first, factor fully, then cancel matching factors (e.g., (x^2−4)/(x−2)=(x−2)(x+2)/(x−2)=x+2 with x≠2; (x+2)/(2x+4)=(x+2)/[2(x+2)]=1/2 with x≠−2).
For addition, make a common denominator and combine numerators-x/(x+2)+1/(x+3)=[x(x+3)+(x+2)]/[(x+2)(x+3)]-then simplify; I think that’s the safest checklist, and this refresher helps: https://www.khanacademy.org/math/algebra/rational-expressions.