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## Homework Statement

Hello! I'm lost on how to start this, I've got formulas given to me from the text, but I have no idea on how to piece everything together. So I need to use established properties of moduli to show that when [tex]\left.\left|z_{3}\right|\neq\left|z_{4}\right|[/tex],

then:

[tex]\left.\frac{Re(z_{1}+z_{2})}{|z_{3}+z_{4}|}\leq\frac{|z_{1}|+|z_{2}|}{||z_{3}|-|z_{4}||}[/tex]

## Homework Equations

Re is the real component of z where z=(x,y) so x = Re z , y= Im z

z can be written as:

z=x+iy

Vector properties

[tex]\left.|z| = \sqrt{x^{2}+y^{2}}[/tex]

[tex]\left.z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})[/tex]

[tex]\left.Re(z) \leq |Re(z)| \leq |z|[/tex]

[tex]\left.|z_{1}\pm z_{2}|\leq|z_{1}|+|z_{2}|[/tex]

[tex]\left.|z_{1}\pm z_{2}|\geq||z_{1}|-|z_{2}||[/tex]

## The Attempt at a Solution

Well what I attempted to do is to say the [tex]\left.Re(z_{1}+z_{2})[/tex] portion can be turned into [tex]\left.x_{1}+x_{2}[/tex].

I can also argue(given the properties) the left denominator is bigger or equal to the right hand side denominator since [tex]\left.|z_{1}\pm z_{2}|\geq||z_{1}|-|z_{2}||[/tex] but it doesn't really help since the left term is suppose to be smaller than the right hand term.

I can also say that the term [tex]\left.|z_{1}|+|z_{2}|[/tex] is given by [tex]\left. \sqrt{x^{2}_{1}+y^{2}_{1}}+\sqrt{x^{2}_{2}+y^{2}_{2}}[/tex] which is bigger than just the [tex]\left.x_{1}+x_{2}[/tex] term?

But these methods are just me trying to prove it going reverse. But I'm just really lost trying to prove it starting with the simple [tex]\left.\left|z_{3}\right|\neq\left|z_{4}\right|[/tex] because the question gives us a result that has 2 addition z terms(where does z1 and z2 come from?)? Please give me some advice on how I can approach this.

Thanks!!!!

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