1
Only like denominators can be added
2/7 + 3/7 = 5/7 because both count sevenths. a/c + b/d counts two different things, and there is no rule that adds them directly — in particular a/c + b/d is not (a + b)/(c + d).
Lesson · Rational and Radical Expressions · Rational expressions
Adding and Subtracting Rational Expressions
Fractions can only be added when they are counted in the same units, so the work is all in the denominator. Building the least one is a factoring problem, and the subtraction that follows is where the sign errors live.
a/c + b/c = (a + b)/c same denominator, add the tops
LCD: every distinct factor, each to its HIGHEST power
(x + 2)(x + 3) and (x + 2)² ⟹ LCD = (x + 2)²(x + 3)
a/c − b/c = (a − b)/c subtract ALL of b, not its first term
One denominator, built from factors
Core ideas
What this lesson establishes
Adding is the only one of the four operations that cannot be done a piece at a time. Both fractions must first be rewritten over one denominator, and that denominator is chosen, not stumbled upon.
2
The LCD is assembled from factors
Factor each denominator, then take every distinct factor to the highest power it reaches in any one of them. The product of the denominators always works, but it is often larger than necessary.
3
Subtraction distributes over the whole numerator
a/c − (b + d)/c is (a − b − d)/c. The bar acts as a bracket, and the moment you write the numerators on one line that bracket has to be written explicitly or it is lost.
The material
Building the denominator, then the numerators
Why a common denominator is required, and how to find the least one.
With a common denominator the rule is immediate: a/c + b/c = (a + b)/c. Everything difficult about this lesson is in arranging for that situation to hold.
Definition · Least common denominator
The LCD of two rational expressions is the product of every distinct factor appearing in either denominator, each raised to the highest power with which it appears in a single denominator.
Example · Building an LCD
For 5/((x + 2)(x + 3)) and 3/(x + 2)² the distinct factors are x + 2 and x + 3. The highest power of x + 2 in one denominator is 2, and of x + 3 is 1, so the LCD is (x + 2)²(x + 3) — not the product of the two denominators, which would carry (x + 2)³.
Multiplying the denominators together is never wrong, only wasteful: the extra factor cancels at the end, after you have carried it through every line. On a two-term sum that is a nuisance; in “Solving Rational Equations”, where the LCD multiplies an entire equation, it is a real cost.
Once the LCD is chosen, each fraction is rebuilt to sit over it. Multiply the numerator and the denominator of each by whatever factors the LCD has and it lacks — legal by the fundamental principle of “Simplifying Rational Expressions”, since you are multiplying by a form of 1.
5/((x + 2)(x + 3)) + 3/(x + 2)² LCD (x + 2)²(x + 3)
= 5(x + 2)/LCD + 3(x + 3)/LCD
= (5x + 10 + 3x + 9)/LCD
= (8x + 19)/((x + 2)²(x + 3))
Now subtraction, and the error that this lesson exists to prevent. When the second numerator has more than one term, the minus sign in front of the fraction applies to all of it.
Example · Where the sign goes
x/(x² − 9) − 2/(x + 3) becomes, over the LCD (x − 3)(x + 3), the numerator x − 2(x − 3). That is x − 2x + 6 = 6 − x. Writing x − 2x − 6 instead — distributing the minus to the first term only — gives −x − 6, and every later step is wrong.
The reliable defence is mechanical: put the entire second numerator in brackets before you subtract, and only then distribute. It costs one pair of brackets and removes the most common source of lost marks in the whole topic.
Finally, the answer's domain is inherited from the denominators you started with, exactly as in “Simplifying Rational Expressions”. (6 − x)/((x − 3)(x + 3)) excludes 3 and −3, and would still exclude them if the numerator had cancelled a factor away.
Interactive
Adding and subtracting
The least common denominator, and the minus sign that hits every term
Method
Adding or subtracting
Five minutes on the denominator saves the numerator.
1
Factor every denominator
The LCD is defined in terms of factors, so it cannot be built from x² − 9. It can be built from (x − 3)(x + 3).
2
Assemble the LCD
Each distinct factor once, to the highest power it reaches in any single denominator. Check it is divisible by each original denominator.
3
Rebuild each fraction over the LCD
Multiply top and bottom by the missing factors. Leave the numerator unexpanded for the moment, in brackets.
4
Combine the numerators, then simplify
Distribute the sign, collect like terms, and factor the result to see whether anything cancels — carrying the original exclusions forward.
Denominators and signs
Score 0 / 0Choose the answer you can justify.
Simplify x/(x² − 9) − 2/(x + 3)
The brackets in line four are the entire point of the example.
x/(x² − 9) − 2/(x + 3)
x² − 9 = (x − 3)(x + 3) LCD = (x − 3)(x + 3)
exclusions: x ≠ 3, x ≠ −3
= x/((x − 3)(x + 3)) − 2(x − 3)/((x − 3)(x + 3))
= [x − (2x − 6)] / ((x − 3)(x + 3))
= (x − 2x + 6) / ((x − 3)(x + 3))
= (6 − x)/((x − 3)(x + 3)), x ≠ 3, x ≠ −3
check at x = 0 : 0 − 2/3 = −2/3
6/((−3)(3)) = 6/(−9) = −2/3 ✓
Note that 6 − x shares no factor with the denominator, so the answer is already in lowest terms. Note also that it is not −1 times something familiar: 6 − x and x − 3 are not opposites, and the sign trick from “Simplifying Rational Expressions” does not apply here.
For a faded subtraction, take 2/(x² − x − 6) − 1/(x − 3). The supplied factorisation is x² − x − 6 = (x − 3)(x + 2), which is also the LCD. Rebuild the second fraction, bracket its new numerator and finish. Check against −x/[(x − 3)(x + 2)], with x ≠ 3, −2; a numerator of 4 − x reveals that the minus sign did not reach the +2.
Common mistakes
1
Subtracting only the first term of the numeratora/c − (b + d)/c is (a − b − d)/c. Bracket the numerator before you subtract and the error becomes impossible rather than merely unlikely.
2
Adding the denominatorsa/c + b/d is not (a + b)/(c + d). One numerical instance settles it for good, and it is worth doing once so the rule never returns.
3
Cancelling a factor of the new denominator against one term on topIn (8x + 19)/((x + 2)²(x + 3)) nothing cancels, because 8x + 19 is not divisible by either factor. A factor must divide the whole numerator, not part of it.
Check your understanding
✓
Finish when the LCD is built from factors and the sign survives.
You should be able to state the LCD of two factored denominators immediately, and to subtract two fractions whose numerators are binomials without dropping a sign. Those are the two skills “Complex Fractions” assumes and “Solving Rational Equations” runs on.
Note: The lab writes the numerator in brackets and pauses there before distributing, because that intermediate line is the one nobody writes and everybody needs. The exclusions shown are those of the original denominators, which is why they can outnumber the factors visible in the answer.