Solve 7-2/b<5/6

$$7- \frac{ 2 }{ b } < \frac{ 5 }{ 6 }$$

Solve for b

$$b\in \left(0,\frac{12}{37}\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{b}{b}.

$$\frac{7b}{b}-\frac{2}{b}<\frac{5}{6}$$

Since \frac{7b}{b} and \frac{2}{b} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7b-2}{b}<\frac{5}{6}$$

Factor out b.

$$\frac{1}{6}\times \frac{1}{b}\left(37b-12\right)<0$$

For the product to be negative, b and b-\frac{12}{37} have to be of the opposite signs. Consider the case when b is positive and b-\frac{12}{37} is negative.

$$b>0 b-\frac{12}{37}<0$$

The solution satisfying both inequalities is b\in \left(0,\frac{12}{37}\right).

$$b\in \left(0,\frac{12}{37}\right)$$

Consider the case when b-\frac{12}{37} is positive and b is negative.

$$b-\frac{12}{37}>0 b<0$$

This is false for any b.

$$b\in \emptyset$$

The final solution is the union of the obtained solutions.

$$b\in \left(0,\frac{12}{37}\right)$$