Question

Solve each inequality. Graph the solution set.

$\frac{2x-6}{1-x}<2$

Short answer

The real zeroes are 1 and 2 and the solution set can be graphed as

Step-by-step solution

Step 1. Given Information 

The given inequality is$\frac{2x-6}{1-x}<2$

Step 2. Calculation

Solve the given inequality as follows,

$$\begin{gathered} \frac{2x-6}{1-x}-2<2-2 \\ \frac{2x-6}{1-x}-2<0 \\ \frac{2x-6}{1-x}-2\left(\frac{1-x}{1-x}\right)<0 \\ \frac{4x-8}{1-x}<0 \end{gathered}$$

Now, let $f\left(x\right)=\frac{4x-8}{1-x}$. We will find the real zeroes of the numerator and denominator.

$$\begin{gathered} 4x-8=0\Rightarrow x=2 \\ 1-x=0\Rightarrow x=1 \end{gathered}$$

Thus, real zeroes of the function are 1 and 2

Step 3. Graph 

Construct the table by using a number from each interval to check whether the function is positive or negative.

Thus, the solution set is $\left\{x|x<-1orx>2\right\}$

And the solution set can be graphed as