Multiplying a postivie number by a absolute expression

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Homework Help Overview

The discussion revolves around the manipulation of expressions involving absolute values, specifically whether a positive coefficient can be multiplied with an absolute expression like 5|x+2|. Participants explore the implications of the sign of the coefficient on the validity of such operations.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants question the conditions under which a coefficient can be multiplied with an absolute expression, particularly focusing on the necessity of the coefficient being positive. Others discuss the definition of modulus and its implications for different cases of the variable.

Discussion Status

The discussion is ongoing, with participants providing insights into the properties of absolute values and the effects of positive and negative coefficients. There is acknowledgment of the need for clarity regarding when it is appropriate to multiply coefficients with absolute expressions.

Contextual Notes

Some participants express confusion about the rules governing the multiplication of coefficients with absolute values, particularly in cases where the coefficient is negative. The discussion includes references to specific examples to illustrate these points.

sgstudent
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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!
 
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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:
[tex]|x+2| = \{\begin{array} ((x+2) \text{ if (x+2) ≥ 0} \\ -(x+2) \text{ if (x+2) ≤ 0} \end{array}[/tex]
[tex]|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|[/tex]
for all k≥0.
 
Last edited by a moderator:
Fightfish said:
Clearly |x+2| is a positive number, as you pointed out.
Or zero.
 
sgstudent said:

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!

[tex]a|b| =\left\{ \begin{array}{rl} |a b| &\text{if } a \geq 0\\<br /> - |ab| & \text{if } a < 0.<br /> \end{array} \right.[/tex]
Note that [itex]|a b| = |(-a) b|.[/itex]

RGV
 
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!
 
sgstudent said:
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.
 
Thanks for the help!
 

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