Answer
The two consecutive numbers are $709$ and $710$.
Work Step by Step
$\bf{\text{l. Familiarize yourself with the problem.}}$
If one apartment is numbered with $x$,
the next one is numbered with $x+1.$
We need the sum of $x$ and $(x+1)$ to be $1419$
$\bf{\text{2. Translate to mathematical language. (This often means writing an equation.)}}$
$x+x+1=1419$
$\bf{\text{3. Carry out some mathematical manipulation. (This often means solving an equation.)}}$
$ 2x+1=1419\qquad$... subtract 1
$ 2x=1418\qquad$... divide with 2
$x=709$.
Then, $x+1=710.$
The two consecutive numbers are $709$ and $710$.
$\bf{\text{4. Check your possible answer in the original problem.}}$
Is the sum $1419$?
Yes.
$\bf{\text{5. State the answer clearly, using a complete sentence.}}$
The two consecutive numbers are $709$ and $710$.
