Question

Solve the inequality:-
$\sf\dfrac{x-1}{x-2}\ \textgreater \ 5$

Answer should be in the form of intervals

Answer 1

ANSWER:

Given:

• (x - 1)/(x - 2) > 5

To Find

• Value of x

Solution:

We are given that,

$:\implies\sf\dfrac{x-1}{x-2} > 5$

Transposing 5 to LHS,

$:\implies\sf\dfrac{x-1}{x-2}-5>0$

Taking LCM,

$:\implies\sf\dfrac{(x-1)-5(x-2)}{x-2} > 0$

So,

$:\implies\sf\dfrac{x-1-5x+10}{x-2} > 0$

On simplifying,

$:\implies\sf\dfrac{-4x+9}{x-2} > 0$

In order to solve this non-linear inequality, we will apply the zero product theorem.

That means that each variable factor in the original inequality needs to be set to zero. These 2 points determine 3 intervals on the number line. After these equations are solved, we can determine the solution intervals of the original inequality.

So,

$:\implies\begin{cases}\sf{-4x+9=0}\\\sf{x-2=0}\end{cases}$

So,

$:\implies\begin{cases}\sf{-4x=-9}\\\sf{x-2=0}\end{cases}$

So,

$:\implies\begin{cases}\sf{4x=9}\\\sf{x=2}\end{cases}$

Hence,

$:\implies\begin{cases}\sf{x=\dfrac{9}{4}}\\\sf{x=2}\end{cases}$

So, now we have found 2 points for the given inequality. Now, we'll plot them on number line.

(Refer Attachment)

We had,

$:\implies\sf\dfrac{-4x+9}{x-2} > 0$

After plotting these 2 points, we'll see which interval gives us answer greater than 0.

So, the interval (2, 9/4) gives us positive values.

Therefore,

$:\implies\bf x\in\left(2,\dfrac{9}{4}\right)$