The inequality is
[tex]7- \frac{2}{b}\ \textless \ \frac{5}{b}[/tex]
write 7 as 7b/b to have all the expressions in common denominator:
[tex] \frac{7b}{b} - \frac{2}{b}\ \textless \ \frac{5}{b}[/tex]
[tex]\frac{7b-2}{b} \ \textless \ \frac{5}{b}[/tex]
[tex]\frac{7b-2}{b}- \frac{5}{b}\ \textless \ 0[/tex]
[tex]\frac{7b-2-5}{b}\ \textless \ 0[/tex]
[tex]\frac{7b-7}{b}\ \textless \ 0[/tex]
[tex]\frac{7(b-1)}{b}\ \textless \ 0[/tex]
here b=1 is a root and b=0 is not in the domain of the expression, but it still has an effect in the sign of the expression.
the sign table of [tex]\frac{7(b-1)}{b} [/tex] is :
+++++++[0] --------[1] +++++
this means that for values of b to the left of 0 and to the right of 1, the expression is positive, and for values of b in (0, 1), the expression is negative.
that is [tex]\frac{7(b-1)}{b}\ \textless \ 0 [/tex] for b∈(0, 1)
Answer: (0, 1)
