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Question Number 156164 by MathSh last updated on 08/Oct/21

Answered by ghimisi last updated on 09/Oct/21

$$\mid x\mid =\frac{1}{2}\mid x+y+x-y\mid ⩽\frac{1}{2}\mid x+y\mid +\frac{1}{2}\mid y-x\mid \left(1\right)$$$$\mid y\mid =\frac{1}{2}\mid y+x+y-x\mid ⩽\frac{1}{2}\mid x+y\mid +\frac{1}{2}\mid y-x\mid \mid \left(2\right)$$$$\mid 3x+2y\mid =\mid 4x+3y-(x+y)\mid ⩽\mid 4x+3y\mid +\mid x+y\mid \left(3\right)$$$$\left(1\right)+\left(2\right)+\left(3\right)\Rightarrow$$$$\mid x\mid +\mid y\mid +\mid 3x+2y\mid ⩽\mid 4x+3y\mid +2\mid x+y\mid +\mid y-x\mid$$

Commented by MathSh last updated on 09/Oct/21

$$\mathrm{Perfect}\mathrm{dear}\mathrm{Ser},\mathrm{thank}\mathrm{you}$$

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