Multiplying a postivie number by a absolute expression

[sgstudent]

Homework Statement

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

Homework Equations

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!

 

[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.

 

[Mark44]

Clearly |x+2| is a positive number, as you pointed out.

Or zero.

 

[Ray Vickson]

Homework Statement

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

Homework Equations

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!

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

RGV

 

[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!

 

[micromass]

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!

Yes, that is correct.

 

[sgstudent]

Thanks for the help!