# How many values are there for the cot(3x-pi/4)!=-1? [-5,5]

Apr 22, 2018

There are infinitely many values of x that will make the cotangent function not equal to -1 within the domain $\left[- 5 , 5\right]$.

#### Explanation:

Given: $\cot \left(3 x - \frac{\pi}{4}\right) \ne - 1 , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\}$

Use the identity $\cot \left(u\right) = \frac{1}{\tan} \left(u\right)$ where $u = 3 x - \frac{\pi}{4}$:

$\frac{1}{\tan} \left(3 x - \frac{\pi}{4}\right) \ne - 1 , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\}$

$\tan \left(3 x - \frac{\pi}{4}\right) \ne \frac{1}{-} 1 , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\}$

$\tan \left(3 x - \frac{\pi}{4}\right) \ne - 1 , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\}$

$3 x - \frac{\pi}{4} \ne {\tan}^{-} 1 \left(- 1\right) , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\}$

$3 x - \frac{\pi}{4} \ne \frac{3 \pi}{4} + n \pi , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\} , n \in \mathbb{Z}$

$3 x \ne \pi + n \pi , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\} , n \in \mathbb{Z}$

$x \ne \frac{\pi}{3} + n \frac{\pi}{3} , \left\{x \in \mathbb{R} | - 5 \le x \le 5\right\} , n \in \mathbb{Z}$

There are infinitely many values of x that make the above inequality true within the domain restrictions.