Trouble understanding absolute value inequality property of integrals

Question

I am having trouble understanding the following property of definite integrals:

$\big|\int_a^b f(x)dx| \leq \int_a^b|f(x)|dx $

1. What would be an example of where the left hand side is less than the right hand side?

2. Also, I was able to come up with a (seeming) violation of this rule when $a > b$

   $\big|\int_5^2 -4dx|= 42 > \int_5^2|-4|dx = -42 $

   Am I supposed to be assuming that $a < b$ ? What am I doing wrong?

Answer 1

Define $f(x)=x$ on [-1, 1].$$ \big|\int_{-1}^{1} xdx|=0 \le \int_{-1}^{1}|x|dx =1$$

Sketching a graph will be quite helpful.