How do you solve and write the following in interval notation: -(x + 6)(x + 2)(x - 2) >0?

Oct 17, 2017

$\left(x < - 6\right)$ or $\left(- 2 < x < 2\right)$

Explanation:

The expression $\textcolor{b l u e}{- \left(x + 6\right) \left(x + 2\right) \left(x - 2\right)}$
is obviously equal to $\textcolor{b l u e}{0}$ at $x \in \left\{\textcolor{m a \ge n t a}{- 6} , \textcolor{m a \ge n t a}{- 2} , \textcolor{m a \ge n t a}{+ 2}\right\}$

This gives us the critical points where the value of the expression might change between positive and negative (where the inequality $< 0$ might apply).

We can select any other values below this set, between values in this set, and above this set to determine if the expression is $> 0$ between these critical values.
For example:
{: ("critical value at "x=,,color(magenta)(-6),,color(magenta)(-2),,color(magenta)(+2),), ("sample value for "x,=color(green)(-7),,color(green)(-4),,color(green)0,,color(green)(+4)) :}

Evaluating at these sample values:
{: ("when "x=," | ",-(x+6)(x+2)(x-2)=," | ","is this "> 0?), (color(green)(-7)," | ",+45," | ",color(red)("Yes")), (color(green)(-4)," | ",-24," | ","No"), (color(white)(.)color(green)0," | ",+24," | ",color(red)("Yes")), (color(green)(+4)," | ",-45," | ","No") :}

This gives us that
for $\textcolor{red}{x < - 6}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{- \left(x + 6\right) \left(x + 2\right) \left(x - 2\right) > 0}$
for $- 6 < x < - 2$
$\textcolor{w h i t e}{\text{XXX}} - \left(x + 6\right) \left(x + 2\right) \left(x - 2\right) \cancel{>} 0$
for $\textcolor{red}{- 2 < x < + 2}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{- \left(x + 6\right) \left(x + 2\right) \left(x - 2\right) > 0}$
for $x > + 2$
$\textcolor{w h i t e}{\text{XXX}} - \left(x + 6\right) \left(x + 2\right) \left(x - 2\right) \cancel{>} 0$

[note that at $x = - 6 \mathmr{and} - 2 \mathmr{and} + 2$ the expression is $= 0$ and
therefore it can not be $> 0$ for these values.