Question

Rewrite without absolute value for the given conditions:
1. y=|x−3|+|x+2|−|x−5|, if 3 2. y=|x−5|+|x+5|, if x>5
3. y=|x−5|+|x+5|, if x<−5
4. y=|x−5|+|x+5|, if −5 Pls help! extra credit!

Answer 1

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,

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