Calculating solubility of iron(III) hydroxide in water
In my version of the book the original assumption (step 7a) is shown to lead to a contradiction when later checked (logically, this form of proof is called reductio ad absurdum or proof by contradiction). The point is that you should 1) learn not to be afraid to apply assumptions, but that you should then 2) verify that the original assumption was indeed acceptable. This is a strategy repeatedly encouraged in the book. 

To show how this leads to a contradiction, 


make the assumption, so that $ \ce{3[Fe^{3+}]} = \ce{[OH-]}$
solve for $S = \ce{[Fe^{3+}]}$  based on the $K_{sp}$ and the assumption
Compute $\ce{[OH-] = \ce{3[Fe^{3+}]}  }$, following the original assumption
Compute $\ce{[H+]}$ from $K_w$ and $\ce{[OH-]}$
Check the accuracy of the assumption 


It is by the way not difficult to solve the problem exactly, by computing the $\ce{[OH-]}$ as follows:

$$ \ce{[OH-]}=\mathrm{\sqrt\frac{K_w+\sqrt{K_w^2 +12K_{sp}}}{2}}$$

and then applying the expression for the solubility constant to compute the solubility.
However the point of the problem is to illustrate how to follow this line of reasoning when solving problems (the use and verification of assumptions).