How do you solve #abs(x+10)=4x-8#?

Question

Isaiah Anthony · Accepted Answer

#x = 6#

Your absolute value equation looks like this

#|x+10| = 4x-8#

Right from the start, you know that the solutions to this equation must satisfy the condition

#4x-8>0 <=> x>2#

That happens because the absolute value of a number, regardlesss if that number is positive or negative, is always positive.

#color(blue)( |a| = {(a",", "if " a>=0), (-a",", "if "a<0):})#

So, with this in mind, determine the equation's two possible solutions

#|x+10| = x+10#

and the equation becomes

#x+10 = 4x - 8 => x = 18/3 = color(green)(6)#

#|x+10| = -(x+10) = -x-10#

This will get you

#-x-10 = 4x+8 => x = color(red)(-18/5)#

This solution, #x=-18/5#, will be an extraneous solution because it does not satisfy the contion #x>2#.

Therefore, #x=6# is the only solution to this equation.

Isaiah Anthony · Answer

undefined To solve the equation abs(x+10) = 4x - 8, follow these steps:

1. Set up two equations based on the definition of absolute value:
   a) x + 10 = 4x - 8
   b) -(x + 10) = 4x - 8

2. Solve equation (a) for x:
   x + 10 = 4x - 8
   10 + 8 = 4x - x
   18 = 3x
   x = 6

3. Solve equation (b) for x:
   -(x + 10) = 4x - 8
   -x - 10 = 4x - 8
   -10 + 8 = 4x + x
   -2 = 5x
   x = -2/5

So, the solutions are x = 6 and x = -2/5.