Question

Solve the inequation $(x+1){(x-3)}^{80}{(x-5)}^7{(x-4)}^{28}{(x-2)}^{79}<0$. Find the greatest integer satisfying the equation.

Answer 1

Given $(x+1){(x-3)}^{80}{(x-5)}^7{(x-4)}^{28}{(x-2)}^{79}<0$

Since, the even power terms will always greater than 0

$\Rightarrow (x+1)(x-5{)}^7(x-2{)}^{79}<0$ ....(2)

So, we have three critcal points $-1,2,5$

Now, taking a value greater than $5$, does not satisfy the inequality (2)

Now, we will take a value in between $2$ and $5$ , which also does not satisfy inequality (2)

Now, we will take a value in between $-1$ and $2$ , which also does not satisfy inequality (2)

Now, we will take any value lesser than $-1$ ,which satisfy the inequality (2).

So, the solution set of inequality (2) is $(-\infty ,-1)$

Hence, the solution set of inequality (1) is $(-\infty ,-1)$

So, the greatest integer satisfying the given inequality is $-2.$