Algebraic fractions

Cancelling an $x$ or a number that is only part of a sum — e.g. cancelling the $x^2$ in $\dfrac{x^2-9}{x^2+7x+12}$ instead of cancelling a whole bracket.

Forgetting that subtracting a fraction means subtracting all of its numerator, e.g. $-(x-2)=-x+2$, not $-x-2$.

When clearing denominators to solve, some terms get multiplied but not others — e.g. forgetting to multiply the constant on the right-hand side by the common denominator.

Adding $\dfrac{2}{x}+\dfrac{3}{x+1}$ by putting each numerator over its own denominator. The common denominator is $x(x+1)$: $\dfrac{2(x+1)+3x}{x(x+1)}$.

After multiplying two fractions, simplify: $\dfrac{3x}{4}\times\dfrac{8}{9x^2}=\dfrac{24x}{36x^2}=\dfrac{2}{3x}$. Leaving $\dfrac{24x}{36x^2}$ is not fully simplified.

Treating $16x^2-36$ as $(2x-3)(2x+3)$ drops the factor of $4$ — it is $4(2x-3)(2x+3)$. Pull out the common factor first (or factor as $(4x-6)(4x+6)$ and then take the $2$ out of each bracket).