In all the places I've seen this property described;
$$\left|{\int_{{\,a}}^{{\,b}}{{f(x)\,\mathrm{d}x}}} \right| \le \int_{{\,a}}^{{\,b}}{{\left| {f\left( x \right)} \right|\,\mathrm{d}x}}$$
The condition that $a \leq x \leq b$ is never specified. I assume there must be such a condition, because considering, say, $y=x^2$ from $x=2$ to $x=1$,(so that $a$ is greater than $b$) the absolute values don't impact the result, and obviously the expression on the right of the inequation is negative, while the other isn't.
You are right, it must be said that $a\le b$, otherwise the inequality is clearly false. (For some strange reason someone downvoted your question, but it is a good question to clarify ideas).
To add something, the property cited follows from the definition of the integral of Riemann. If we use the integral of Lebesgue then the notation used doesn't lead to any confusion about $a\le b$ because we just write $\int_{[c,d]}f $ instead of $\int_c^d f$. For the integral of Lebesgue the property holds as well.