The question:
Let $f$ be continuous over the reals such that $\int_{-\infty}^{\infty}|f(x)|dx$ exists. Show that $\int_{-\infty}^{\infty}f(x)dx$ exists.
Would it be sufficient to go about this using the Triangle Inequality (i.e. $f(x) \leq |f(x)|$ over the entire integral?)
Not sure as to why this was marked off-topic. This is an important question about improper integration, and I presented my line of thought above. I urge those who find it off-topic to demonstrate why, as this is exactly how the question was presented to me.
Almost. What you want to show is that $$ \lim_{N\to\infty}\int_N^\infty f(x)\,dx=0 $$ (and the analog case at $-\infty$). You achieve that by $$ \left|\int_N^\infty f(x)\,dx\right|\leq\int_N^\infty|f(x)\,dx. $$