Question

solution set of the inequation x²/x-2>0 is: ​

Answer 1

We're asked to find solution set of the inequality,

$\displaystyle\sf{\longrightarrow \dfrac {x^2}{x-2}>0}$

We know the denominator should be non - zero.

$\displaystyle\sf{\longrightarrow x-2\neq0}$

$\displaystyle\sf{\longrightarrow x\neq2}$

$\displaystyle\sf{\longrightarrow x\in\mathbb{R}-\{2\}\quad\quad\dots(1)}$

The numerator $\displaystyle\sf {x^2}$ is always non - negative $\displaystyle\sf {\forall x\in\mathbb{R}.}$

But since the fraction is only greater than 0,

$\displaystyle\sf{\longrightarrow x^2>0}$

$\displaystyle\sf{\Longrightarrow x\in\mathbb{R}-\{0\}\quad\quad\dots(2)}$

For a fraction being positive, both the numerator and denominator should have same sign.

Hence denominator is also positive here.

$\displaystyle\sf{\longrightarrow x-2>0}$

$\displaystyle\sf{\longrightarrow x>2}$

$\displaystyle\sf{\Longrightarrow x\in(2,\ \infty)\quad\quad\dots(3)}$

Now we take (1), (2) and (3) in common.

Then we get,

$\displaystyle\sf{\longrightarrow\underline {\underline {x\in(2,\ \infty)}}}$

So this is the solution.