# How do you solve the inequality 1+5/(a-1)<=7/6?

Apr 11, 2018

color(blue)((-oo,1)uu[31,oo)

#### Explanation:

$1 + \frac{5}{a - 1} \le \frac{7}{6}$

Subtract $\frac{7}{6}$ form both sides:

$\frac{5}{a - 1} - \frac{1}{6} \le 0$

$\frac{30 - \left(a - 1\right)}{6 \left(a - 1\right)} \le 0$

$\frac{31 - a}{6 \left(a - 1\right)} \le 0$

Multiply by $6$:

$\frac{31 - a}{a - 1} \le 0$

So we need:

$\frac{+}{-}$ and $\frac{-}{+}$

$31 - a \ge 0 \implies a \le 31$

$a - 1 < 0 \implies a < 1$

$31 - a \le 0 \implies a \ge 31$

$a - 1 > 0 \implies a > 1$

$a < 1$ and $a \ge 31$

color(blue)((-oo,1)uu[31,oo)