Question

Solve the inequality. Check your solutions.

Answer 1

The inequality is 

$7- \frac{2}{b}\ \textless \ \frac{5}{b}$

write 7 as 7b/b to have all the expressions in common denominator:

$\frac{7b}{b} - \frac{2}{b}\ \textless \ \frac{5}{b}$

$\frac{7b-2}{b} \ \textless \ \frac{5}{b}$

$\frac{7b-2}{b}- \frac{5}{b}\ \textless \ 0$

$\frac{7b-2-5}{b}\ \textless \ 0$

$\frac{7b-7}{b}\ \textless \ 0$

$\frac{7(b-1)}{b}\ \textless \ 0$

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 $\frac{7(b-1)}{b}$ 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 $\frac{7(b-1)}{b}\ \textless \ 0$ for b∈(0, 1)

Answer: (0, 1)