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$\longrightarrow \large \tt{Verify \: that \: : |x + y| < |x| + |y| \: For \: x = \frac{ - 5}{12} \: and \: y = \frac{ - 7}{18}}$

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Answer 1

Corrected question:

Verify that | x + y | = | x | + | y |, for x = (-5)/(12) and y = (-7)/(18)

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Solution:

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Value of | x + y | (Absolute value of x + y):

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$\sf \dashrightarrow \left| \ x + y \ \right|$

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Substitute the value of 'x' and 'y'.

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$\sf \dashrightarrow \left| \ \left( \dfrac{-5}{12} \right) + \left( \dfrac{-7}{18} \right) \ \right|$

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The LCM of 12 and 18 is 36, on taking LCM we get;

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$\sf \dashrightarrow \left| \ \left( \dfrac{-5}{12} \times \dfrac{3}{3} \right) + \left( \dfrac{-7}{18} \times \dfrac{2}{2} \right) \ \right|$

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$\sf \dashrightarrow \left| \ \left( \dfrac{-15}{36} \right) + \left( \dfrac{-14}{36} \right) \ \right|$

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$\sf \dashrightarrow \left| \ \left( \dfrac{-15 - 14}{36} \right) \ \right|$

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$\sf \dashrightarrow \left| \ \dfrac{\textsf{\textbf{-29}}}{\textsf{\textbf{36}}} \ \right|$

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The absolute value of a negative fraction will be positive, therefore;

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$\sf \dashrightarrow \ \dfrac{\textsf{\textbf{29}}}{\textsf{\textbf{36}}} \ \dots \ Relation(1)$

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Value of | x | + | y | (Absolute value of x + Absolute value of y):

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$\sf \dashrightarrow \left| \ x \ \right| + \left| \ y \ \right|$

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$\sf \dashrightarrow \left| \ \dfrac{-5}{12} \ \right| + \left| \ \dfrac{-7}{18} \ \right|$

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We know that the absolute value of a negative number is positive, therefore;

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$\sf \dashrightarrow \dfrac{5}{12} + \dfrac{7}{18}$

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The LCM of 12 and 18 is 36, on taking LCM we get;

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$\sf \dashrightarrow \left(\dfrac{5}{12} \times \dfrac{3}{3} \right) + \left(\dfrac{7}{18} \times \dfrac{2}{2} \right)$

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$\sf \dashrightarrow \dfrac{15}{36} + \dfrac{14}{36}$

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$\sf \dashrightarrow \dfrac{15 + 14}{36}$

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$\sf \dashrightarrow \dfrac{\textsf{\textbf{29}}}{\textsf{\textbf{36}}} \ \dots Relation(2)$

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We've been asked to verify that;

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$\sf \dashrightarrow | x + y | = | x | + | y |$

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Substitute the values we've gotten in Relation(1) and Relation(2).

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$\sf \dashrightarrow \dfrac{29}{36} = \dfrac{29}{36}$

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$\sf \dashrightarrow LHS = RHS$

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Therefore, verified.