Question

PROVE: Triangle Inequality We prove Property 5 of abso- lute values, the Triangle Inequality: $$|x+y| \leq|x|+|y|$$ (a) Verify that the Triangle Inequality holds for $x=2$ and $y=3,$ for $x=-2$ and $y=-3,$ and for $x=-2$ and $y=3 .$ (b) Prove that the Triangle Inequality is true for all real num- bers $x$ and $y .$ [Hint: Take cases. $]$

          PROVE: Triangle Inequality We prove Property 5 of abso-
lute values, the Triangle Inequality:
$$|x+y| \leq|x|+|y|$$
(a) Verify that the Triangle Inequality holds for $x=2$ and
$y=3,$ for $x=-2$ and $y=-3,$ and for $x=-2$ and
$y=3 .$
(b) Prove that the Triangle Inequality is true for all real num-
bers $x$ and $y .$ [Hint: Take cases. $]$

Added by Ryan L.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Manisha Sarker

Step 1

For $x=2$ and $y=3,$ we have: $|x+y| = |2+3| = 5$ $|x|+|y| = |2|+|3| = 5$ Since $5 \leq 5,$ the inequality holds. For $x=-2$ and $y=-3,$ we have: $|x+y| = |-2-3| = 5$ $|x|+|y| = |-2|+|-3| = 5$ Again, $5 \leq 5,$ so the inequality holds. For Show more…

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PROVE: Triangle Inequality We prove Property 5 of abso- lute values, the Triangle Inequality: $$|x+y| \leq|x|+|y|$$ (a) Verify that the Triangle Inequality holds for $x=2$ and $y=3,$ for $x=-2$ and $y=-3,$ and for $x=-2$ and $y=3 .$ (b) Prove that the Triangle Inequality is true for all real num- bers $x$ and $y .$ [Hint: Take cases. $]$