Same rules as ordinary fractions. The only new work is factoring the denominators first.
Simplify
3x + 2 + 5x − 3
The common denominator is (x + 2)(x − 3). Each numerator picks up the other bracket: 3(x − 3) + 5(x + 2) = 3x − 9 + 5x + 10 = 8x + 1. D just adds the numerators — the mistake that treats denominators like they cancel.
Exactly as with ⅓ + ⅕ — you cannot add until both are in fifteenths.
3x+2 + 5x−3 is not 82x−1. The same instinct that would give ⅓ + ⅕ = 28 is at work, and it is wrong in both places.
LCD = (x + 2)(x − 3)
nothing in common, so both factors
3(x − 3)(x+2)(x−3) + 5(x + 2)(x+2)(x−3)
each numerator gains the other denominator
3x − 9 + 5x + 10
expand the top
8x + 1(x + 2)(x − 3)
collect; nothing cancels
Leave the denominator factored. It is easier to check for cancelling, and it shows the restrictions at a glance — here x ≠ −2 and x ≠ 3.
Find the LCD of x² − 9 and x + 3.
x² − 9 = (x − 3)(x + 3)
difference of squares
the other denominator is (x + 3)
already inside the first
LCD = (x − 3)(x + 3)
no new factor needed
Multiplying the two denominators together would have given (x−3)(x+3)² — correct but twice the work, and it would need cancelling at the end.
| To build an LCD | Rule |
|---|---|
| every factor that appears | include it once |
| a repeated factor | take its highest power |
| number coefficients | take their LCM |
Simplify 2xx² − 9 − 1x + 3, using the LCD from step 3.
2x(x−3)(x+3) − 1(x − 3)(x−3)(x+3)
the second gains (x − 3)
2x − (x − 3)
bracket it before subtracting
= 2x − x + 3 = x + 3
both signs flip
x + 3(x−3)(x+3)
and now it cancels
= 1x − 3
x ≠ ±3 still
Writing the numerator without brackets loses the sign on the 3, and the fraction then refuses to simplify. If your answer will not cancel, check that subtraction first.
The final expression 1x − 3 looks happy at x = −3, but the question it came from was undefined there. The excluded values come from the denominators you started with.
One more trick worth having: x − 3 and 3 − x differ only by a sign, so 13 − x = −1x − 3. Factoring out −1 makes two denominators match that looked different.
Simplify x + 2 − 3x + 1.
(x + 2)(x + 1)x + 1 − 3x + 1
x + 2 is over 1: it gains (x + 1)
x² + 3x + 2 − 3 = x² + 3x − 1
expand, then subtract
x² + 3x − 1x + 1
x = 0: 2 − 3 = −1 ✓
Different: a term with no fraction bar has denominator 1, so the whole of it is multiplied by the LCD.
Dropping x + 2 straight onto the top gives x + 2 − 3x + 1. At x = 0 that is −1 — but at x = 1 it is 0, while the original is 3 − 1.5 = 1.5.
Simplify 56x − 14x².
LCD = 12x²
LCM of 6 and 4; highest power of x
5 · 2x12x² − 1 · 312x²
× 2x and × 3
10x − 312x²
x = 1: 56 − 14 = 712 ✓
Different: the number part uses the LCM (12, not 24) and the letter part the highest power (x², not x³).
Simplify 2x² + 2x + 1 − 1x + 1.
x² + 2x + 1 = (x + 1)², so LCD = (x + 1)²
the repeated factor, highest power
2 − (x + 1)(x + 1)²
the second gains one (x + 1); bracket it
1 − x(x + 1)², x ≠ −1
x = 0: 2 − 1 = 1 ✓
Different: (x + 1) already sits inside (x + 1)², so the LCD is not (x + 1)³.
When one denominator is the other turned round, the LCD is just one of them. Flip the sign of the second fraction and add. Simplify x² + 4x − 2 + 4x2 − x.
4x2 − x = −4xx − 2
2 − x = −(x − 2), so the sign moves up
x² + 4 − 4xx − 2
one denominator now
= (x − 2)²x − 2
a perfect square on top
= x − 2, x ≠ 2
x = 0: 4−2 + 0 = −2 ✓
Adding straight across gives x² + 4x + 4x − 2, which does not simplify. When an answer refuses to cancel in a question that clearly expects it to, look for a denominator written backwards.
Addition and subtraction items are built so the final numerator cancels with the LCD — the designers want a neat answer like x − 2 or 1. Factor every denominator, flip any backwards bracket, bracket the numerator you subtract, and test x = 1 (or x = 0) in the original and in your answer.
x = 1 breaks nothing, so the original and the answer must agree there.
Simplify
4x − 1 + 2x + 5
Cross-multiply over the common denominator: 4(x + 5) + 2(x − 1) = 4x + 20 + 2x − 2 = 6x + 18, which factors as 6(x + 3). The sign trap is the second bracket — 2(x − 1) is −2, not +2; reading it as +2 gives B, 6x + 22. A just adds the numerators, and C multiplies each numerator by its own bracket instead of the other one: 4(x − 1) + 2(x + 5) = 6x + 6.
