I was looking at this problem: Norm of the operator $Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt$ and was confused about the step with the integral inequality:
$$\left|\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt\right|\\ \leq \int_{-1}^0|f(t)|\ dt+\int_{0}^1|f(t)|\ dt\\$$
Is there a rule for this inequality that I'm missing? In the first equation, if it was addition, I could use the triangle inequality. Here, is it using the assumption that the sum would be greater than the difference?
Notice that you can see the first equation as an addition:
$$\begin{align} \left|\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt\right|&=\left|\int_{-1}^0f(t)\ dt+\left(-\int_{0}^1f(t)\ dt\right)\right|\\\\ &\leq \left|\int_{-1}^0f(t)\ dt\right|+\left|-\int_{0}^1f(t)\ dt\right|\\\\ &= \left|\int_{-1}^0f(t)\ dt\right|+\left|\int_{0}^1f(t)\ dt\right|\\ \\ &\leq \int_{-1}^0|f(t)|\ dt+\int_{0}^1|f(t)|\ dt\\\end{align}$$