# Question #f4b24

Feb 13, 2017

$3 < x < 6$ or $- 2 < x < - 1$.

#### Explanation:

First, $x \ne 0$ since we need the denominator to be non-zero. Assuming that $x > 0$, we can multiply by $x$ without switching the inequality:

$x < {x}^{2} - 6 < 5 x \implies x < {x}^{2} - 6$ and ${x}^{2} - 6 < 5 x$.

The first one can be rewritten as so, by subtracting $x$:

$0 < {x}^{2} - x - 6 \implies 0 < \left(x - 3\right) \left(x + 2\right) \implies x > 3$ or $x < - 2$.

However, we took $x > 0$, so only $x > 3$ is accepted.

The second inequality, is written as:

${x}^{2} - 5 x - 6 < 0 \implies \left(x - 6\right) \left(x + 1\right) < 0 \implies - 1 < x < 6$

Once again, since $x > 0$, we accept $0 < x < 6$.

If $x > 3$ and $0 < x < 6$, then $3 < x < 6$.

Now, let $x < 0$. Then $x > {x}^{2} - 6 > 5 x$, since we multiplied by a negative number ($x$), so the inequality is now the other way around. We can solve them the same way, remembering now that the $>$ changes into $<$ and the $<$ into a $>$. We arrive at the results:

$0 > {x}^{2} - x - 6 \implies 0 > \left(x - 3\right) \left(x + 2\right) \implies - 2 < x < 3$

This time, $x < 0$ so we accept $- 2 < x < 0$.

The second inequality is:

${x}^{2} - 5 x - 6 > 0 \implies \left(x - 6\right) \left(x + 1\right) > 0 \implies x > 6$ or $x < - 1$

Since $x < 0$, we accept $x < - 1$.

Now, we can combine the inequalities:

$- 2 < x < 3$ and $x < - 1$ means $- 2 < x < - 1$

So in the end:

$3 < x < 6$ or $- 2 < x < - 1$.

Or in interval notation:

$x \in \left(- 2 , - 1\right) \cup \left(3 , 6\right)$

Note that you don't have to factor the quadratic polynomials there to solve the quadratic inequality. Simply use the quadratic formula to find the roots, and then if ${r}_{1} , {r}_{2}$ are these roots, (${r}_{1} < {r}_{2}$) then:

The polynomial is positive, if $x > {r}_{2}$ or $x < {r}_{1}$
The polynomial is negative, if ${r}_{1} < x < {r}_{2}$

If the coefficient of ${x}^{2}$ is negative, the above are reversed

${r}_{1 , 2} = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
where $a , b , c$ the coefficients of the polynomial from highest to lowest degree.