Mathematico | Mathematico

The solution to the inequality

x + 8 < 6 \sqrt{x}

is of the form $α < x < β$ .

Find the value of $α β$ .

Hint

Rearrange into the form

x - 6 \sqrt{x} + 8 < 0

Hint

This is

{(\sqrt{x})}^{2} - 6 \sqrt{x} + 8 < 0

so let $u = \sqrt{x}$

Hint

Sketch

y = u^{2} - 6 u + 8

and look for parts of the parabola below the $x$ axis

Solution

We rearrange, then let $u = \sqrt{x}$

\begin{aligned} x + 8 & < 6 \sqrt{x} \\ x - 6 \sqrt{x} + 8 & < 0 \\ u^{2} - 6 u + 8 & < 0 \\ (u - 4) (u - 2) & < 0 \end{aligned}

We make a sketch

parabola sketch

We find that

\begin{aligned} 2 < u & < 4 \\ 2 < \sqrt{x} & < 4 \\ 4 < x & < 16 \end{aligned}