I have a question about distribution and absolute values. I was solving a problem and was wondering if it would be okay to distribute a number into an absolute value with two terms. For example $3|2x+3|+3x^2-5$, is it okay to distribute the $3$ into $|2x+3|$ to get $|6x+9|$? I have searched this on Google but people said it was not okay to do it, but someone else said it was okay to do it. I am really confused, could someone clear this up for me?
2 Answers
If the multiplier is non-negative, it's OK. For example, $2|x| = |2x|$ but $-3|x|=-|3x|$.
There is a definite fact to use here, namely that $|a\cdot b| = |a|\cdot|b|$ for any two real numbers. Since $3>0$, we have:
$3\cdot|2x+3| = |3|\cdot|(2x+3)| = |(3)\cdot(2x+3)|$
Distributing the $3$ gives us $|6x+9|$.
