For what values of b does the value of the fraction 5−2b /4 belong to the interval [−2; 1]?

The easiest way to solve this is as an inequality. Here's what it's saying we have:
[tex]-2 \leq {\frac{5-2b}{4}}\leq1[/tex]

First, multiply everything by 4 to clear the denominator:
[tex]-8\leq5-2b\leq4[/tex]

Subtract 5 from both sides:
[tex]-13\leq-2b\leq-1[/tex]

In the last step, we need to divide everything by -2 which will flip both inequality signs, so we have:
[tex]\frac{13}{2}\geq b \geq \frac{1}{2}[/tex]

So b is in the interval [tex][\frac{1}{2},\frac{13}{2}][/tex].