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Prove the triangle inequality, which stales that if x and
Chapter 8, Problem 7E(choose chapter or problem)
Prove the triangle inequality, which stales that if x and y are real numbers, then |x| + |y| ? |x + y| (where | x | represents the absolute value of x, which equals x if x ? 0 and equals –x if x<0).
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QUESTION:
Prove the triangle inequality, which stales that if x and y are real numbers, then |x| + |y| ? |x + y| (where | x | represents the absolute value of x, which equals x if x ? 0 and equals –x if x<0).
ANSWER:SolutionStep 1Proof by Direct MethodLet us assume that in the first case x + y 0 and second case x + y < 0 where m and n are the real numbers.Absolute Value functionIt is a function which have maximum as well as minimum value. It contain positive and negative number where positive number consider as maximum number and negative number consider as minimum number. |x| = -x < |x| < xNow we use the definition of absolute value function |x + y| = x + y only if when x + y 0 where x + y consider as maximum value.