I’ve seen the full proof of the Triangle Inequality

|x+y|≤|x|+|y|.

However, I haven’t seen the proof of the reverse triangle inequality:

||x|−|y||≤|x−y|.

Would you please prove this using only the Triangle Inequality above?Thank you very much.

**Answer**

|x|+|y−x|≥|x+y−x|=|y|

|y|+|x−y|≥|y+x−y|=|x|

Move |x| to the right hand side in the first inequality and |y| to the right hand side in the second inequality. We get

|y−x|≥|y|−|x|

|x−y|≥|x|−|y|.

From absolute value properties, we know that |y−x|=|x−y|, and if t≥a and t≥−a then t≥|a|.

Combining these two facts together, we get the reverse triangle inequality:

|x−y|≥||x|−|y||.

**Attribution***Source : Link , Question Author : Anonymous , Answer Author : J. W. Tanner*