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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?

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2 Answers

up vote 2 down vote accepted

If the multiplier is non-negative, it's OK. For example, $2|x| = |2x|$ but $-3|x|=-|3x|$.

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There is a definite fact to use here, namely that |a*b| = |a|*|b| for any two real numbers.

So for 3|2x+3| since 3>0 we have : 3*|2x+3| = |3||(2x+3)| = |(3)(2x+3)| (using above fact) which is then |6x+9| just by multiplying out the inside.

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