What is the source of my confusion about inequality?

[Miike012]

I posted the paragraph with the inequality where I am confused...

I also added a portion i highlighted in red

 

[Curious3141]

I posted the paragraph with the inequality where I am confused...

I also added a portion i highlighted in red

The text in the image is not very clear. One of the 'I's appears to have a line on the bottom, the other has the line on top.

Regardless, the portion in the red box looks nonsensical. Everything before that makes sense.

Let the s integral be represented by S, and the t integral be represented by T. Let's call the first I with the bar on the bottom A and the second I with the bar on top B.

So the inequalities are:

S \leq A \leq T ---(1)

and S \leq B \leq T ---(2)

Take (-1) times inequality 1. Remember to reverse all the signs:

-S \geq -A \geq -T

Reverse the order of the terms (just write it backward):

-T \leq -A \leq -S ---(3)

Add (3) to (2):

S-T \leq B-A \leq T-S

and the latter half of that is what's given in the attachment (before the red box).

What's in the red box looks wrong as I said - was that in the original?