What is the solution to 3|x-4| - |2x+3| + |x+5| = 7?

Answer 1

3|x-4| - |2x+3| + |x+5| = 7

To solve for the variable x in |ax + b| = c, solve both ax + b = c and ax + b = –c.

There are two cases to find the absolute value one is positive case and the other is negative case.

ie., 3|x-4| - |2x+3| + |x+5| = 7 and 3|x-4| - |2x+3| + |x+5| = -7

Case1:

3(x-4) - (2x+3) + (x+5) = 7

3x - 12 - 2x - 3 + x + 5 = 7

3x - 2x + x -12 -3 + 5 = 7

2x - 10 = 7

2x = 7+10

2x = 17

x = 17 / 2.

Case 2:

Definition for absolute value is | a | = a if a ≥ 0 and | a | = -a if a < 0

3|x-4| - |2x+3| + |x+5| = -7

Here to find values of x equate to 0,

x-4=0 => x =4>0 so |x-4| is positive.               [ Since | a |= a if a ≥ 0 ]                                       

2x+3=0 => x =-3/2 <0 ,

x+5=0 => x = -5<0

|2x+3| and |x+5| < 0 so they are negative.      [ Since | a |= -a if a < 0 ]

Remove mods,

+3(x-4) - (-(2x+3)) - (x+5) = -7

3x - 12 + (2x + 3) - x - 5 = -7

3x - 12 + 2x + 3 - x - 5 = -7

4x - 14 = -7

4x = -7 + 14

4x = 7

x = 7/4.

Therefore x = 17/2 and x = 7/4.