(NFQ1) Hello! I just want to ask for help whether the answers in the given pictures are correct. If it's not, please help me recheck and resolve it. Please refer to the given pictures below for the answers and questions. Please read the instructions and directions very carefully. Double and triple check your answers, previous tutors got it wrong. PROBLEM/SITUATION:1.) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute value |x +y| of their sum is the sum |x| + |y| of their absolute values. For no.1:Recall that |x| = x iff x >= 0Show ab < 0 leads to a contradiction.Then assume ab>=0.Prove by cases:(i) Let ab=0.(ii) Let ab > 0 NOTE: Type only your answers. Please do not handwritten your answers. Make sure your formulas, solutions and answers' format are all correct. Step 1We have to prove that product xy of two realnumbers x and y is nonnegative if and only if theabsolute value |x + y of their sum is the sum |x| + |y|of their absolute valuesStep 2If part :-Let product xy of two real numbers x and y isnonnegative then we have to prove that absolutevalue x + y of their sum is the sum |x| + |y| of theirabsolute valuesBy triangle inequality|x + y ≤ |x|+|yl|Equality hods if both x and y is positive or both xand y is negative or x = 0 or y = 0Since,xy is nonnegativeSo,both x and y is positive or both x and y is negativeor x = 0 ory=0Hence,|x + y = |x|+|y|Step 3Only if part :-Let |x + y = |x|+|y| then we have to prove that xy isnonnegative(x + y)² = (x + y)²x² + 2xy + y² = x² +2|x||y| + y² = x² + 2|xy| + y²xy = |xy|xy ≥ 0So,xy is nonnegativeHence,Product xy of two real numbers x and y isnonnegative if and only if the absolute value|x + y of their sum is the sum |x| + |y| of theirabsolute values

Answered: PROBLEM/SITUATION: 1.) Prove that the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.6: Inequalities

Problem 80E

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(NFQ1)

Hello! I just want to ask for help whether the answers in the given pictures are correct. If it's not, please help me recheck and resolve it. Please refer to the given pictures below for the answers and questions. Please read the instructions and directions very carefully. Double and triple check your answers, previous tutors got it wrong.

PROBLEM/SITUATION:

1.) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute value |x +y| of their sum is the sum |x| + |y| of their absolute values.

For no.1:

Recall that |x| = x iff x >= 0

Show ab < 0 leads to a contradiction.

Then assume ab>=0.

Prove by cases:

(i) Let ab=0.

(ii) Let ab > 0

NOTE: Type only your answers. Please do not handwritten your answers. Make sure your formulas, solutions and answers' format are all correct.