1. ## simple absolute values

how do you rewrite |3x+4|-|8-16| without absolute values? thanks!

2. You cannot remove absolute values unless you know the specific value. Thus, $\displaystyle |3x+4|$ cannot be simplified.

3. Originally Posted by asnxbbyx113
how do you rewrite |3x+4|-|8-16| without absolute values? thanks!
first, let's find the turning point (i think that's what it's called). where does 3x + 4 become negative? for $\displaystyle x \in (- \infty, -4/3)$, of course. ok. now we have to split the function in 2.

$\displaystyle |3x + 4| - |8 - 16| = |3x + 4|-8$

now because of the absolute values, we need to account for when the function is negative and when it is positive. we already know where it is negative.

so, if $\displaystyle f(x) = |3x + 4| - 8$

then, $\displaystyle f(x) = \left \{ \begin {array}{cc} 3x + 4 - 8, & \mbox { if } x \ge - \frac 43 \\ -(3x + 4) - 8, & \mbox { if } x < - \frac 43 \end{array}\right.$

$\displaystyle \Rightarrow f(x) = \left \{ \begin {array}{cc} 3x -4, & \mbox { if } x \ge - \frac 43 \\ -3x -12, & \mbox { if } x < - \frac 43 \end{array}\right.$