Answers

1. Answer:

   1. y=|x−3|+|x+2|−|x−5|, if -2<x< 3  = x

   2. y=|x−5|+|x+5|, if x>5 = 2x

   3. y=|x−5|+|x+5|, if x<−5 = -2x

   4. y=|x−5|+|x+5|, if −5<x<5 = 10

   Modulus value means it always gives positive value. Therefore,

   Step-by-step explanation:

   if x < a, then |x – a| is positive when |x – a | = a -x, which is positive as a is larger than x.

   and if x >a, then |x-a| is positive when |x-a| = x – a, which is positive as a is smaller than x.

   Applying the same,

   (1) for  -2 < x < 3,

   |x – 3 | = 3 – x, as x <3

   | x + 2 | = | x – (-2)| = x – (-2) = x+2 as x> -2

   | x – 5 | = 5 -x , as x < 5

   therefore, y = |x−3|+|x+2|−|x−5| = 3 – x + x + 2 – (5 – x) = x

   (2) for x > 5,

   |x – 5|= x – 5

   |x  + 5| = | x – (-5)| = x + 5

   therefore y = x – 5 + x + 5 = 2x

   (3) for x < -5,

   |x – 5| = 5 – x

   |x + 5| = |x – (-5)| = -5-x

   therefore y = 5 – x + -5 -x = -2x

   (4) for -5< x < 5,

   |x -5| = 5 – x

   |x +5| = x + 5

   y = 5 – x + x + 5 = 10

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