# Multiplying a postivie number by a absolute expression

1. May 8, 2012

### sgstudent

1. The problem statement, all variables and given/known data

If I have a expression like 5lx+2l, can I multiply the 5 with x+2?
2. Relevant equations

3. The attempt at a solution
I'm guessing that I can only do this if the number in front of the absolute expression is a positive number. Because if -5lx+2l=-10 then multiplying the -5 does not make sense as it would have to be + for all values of the whole expression. So I'm pretty confused about this. Thanks for the help!

2. May 8, 2012

### Fightfish

Clearly |x+2| is a positive number, as you pointed out.
Yes, you can bringing the 5 into the expression without any loss of generality, because it is clear that the positive constant affects all terms equally.
To be rigorous, let us look at the definition of modulus:
$$|x+2| = \{\begin{array} ((x+2) \text{ if (x+2) ≥ 0} \\ -(x+2) \text{ if (x+2) ≤ 0} \end{array}$$
$$|kx+2k| = \{\begin{array} ((kx+2k) \text{ if (kx+2k) ≥ 0} \\ -(kx+2k) \text{ if (kx+2k) ≤ 0} \end{array} = \{\begin{array} (k(x+2) \text{ if (x+2) ≥ 0} \\ -k(x+2) \text{ if (x+2) ≤ 0} \end{array} = k |x+2|$$
for all k≥0.

Last edited by a moderator: May 30, 2012
3. May 8, 2012

Or zero.

4. May 8, 2012

### Ray Vickson

$$a|b| =\left\{ \begin{array}{rl} |a b| &\text{if } a \geq 0\\ - |ab| & \text{if } a < 0. \end{array} \right.$$
Note that $|a b| = |(-a) b|.$

RGV

5. May 30, 2012

### sgstudent

Oh so if I have a this: -5lx+2l=-10, then I can only bring in positive numbers? So -l5x+10l=-10? Thanks for the help guys!

6. May 30, 2012

### micromass

Staff Emeritus
Yes, that is correct.

7. May 31, 2012

### sgstudent

Thanks for the help!