# How do you solve and graph abs(m-2)<8?

Jun 13, 2017

$f \left(m\right) = \left\{\left(m - 10 \textcolor{w h i t e}{\text{xxx") {m:m>=2}),(-m-6color(white)("xxx}} \left\{m : m < 2\right\}\right)\right.$

for $\mathrm{do} m f \in \left(- \infty , 0\right)$

#### Explanation:

Recall that for a modulus function $f \left(x\right) = | x |$, $f \left(x\right) = x$ for $x \ge 0$ and $f \left(x\right) = - x$ for $x < 0$.

$| m - 2 | < 8$
$\therefore | m - 2 | - 8 < 0 |$

If we call this a function

$f \left(m\right) = | m - 2 | - 8$ for $\mathrm{do} m f \in \left(- \infty , 0\right)$

then we can define it as a hybrid function, which follows the standard transformations of a function, we get:

$f \left(m\right) = \left\{\left(\left(m - 2\right) - 8 \textcolor{w h i t e}{\text{xxx") {m:m>=2}),(-(m-2)-8color(white)("xxx}} \left\{m : m < 2\right\}\right)\right.$
$\therefore f \left(m\right) = \left\{\left(m - 10 \textcolor{w h i t e}{\text{xxx") {m:m>=2}),(-m-6color(white)("xxx}} \left\{m : m < 2\right\}\right)\right.$

Of course, given the domain, we need only consider the function $g \left(m\right) = - m - 6$, so to sketch, simply draw a straight line from the point $\left(0 , - 6\right)$ remembering to leave an open circle at that coordinate.