91Ó°ÊÓ

The given inequality is :

$\frac{x+1}{x-1}>0$

We have to solve for x.

Zeroes of the inequality $f\left(x\right)=\frac{x+1}{x-1}>0$are

$\frac{x+1}{x-1}=0$

$f\left(x\right)$is undefined for $x=1$

Therefore$x=-1$

Now we use the zeroes to separate the real number line into intervals$(-∞,-1),(-1,1),(1,∞)$

Now we select a test number in each interval found in Step 3 and evaluate at each number to determine if

$f\left(x\right)=\frac{x+1}{x-1}=0$is positive or negative

In the interval $(-∞,-1)$, we chose -2 where f is positive

In the interval $(-1,1)$, we chose 0 where f is negative

In the interval $(1,∞)$, we chose 2 where f is positive

We know that our required inequality is $f\left(x\right)>0$

Here the inequality is not strict $(â‰\yen or≤)$so we have to exclude the solutions of $f\left(x\right)=0$in the solution set.

So we want the interval where f is positive.

So required solution set is$(-∞,-1)∪(1,∞)$