How do you solve #abs(3x+5)-2x=3x+4#?

1 Answer
Aug 2, 2015

Answer:

#x = 1/2#

Explanation:

Start by isolating the modulus on one side of the equation

#|3x+5| - color(red)(cancel(color(black)(2x))) + color(red)(cancel(color(black)(2x))) = 3x + 4 + 2x#

#|3x+5| = 5x+4#

If you take into account the fact that the absolute value of a number, regardless if that number is positive or negative, is always positive

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

then you can say that the solutions to this equation must satisfy the condition

#5x+4>0 <=> x > -4/5#

Now, this equation can produce two solutions, depending on which condition is true

  • If #(3x+5)>0#, you have

#|3x + 5| = 3x+5#

and the equation becomes

#3x+5 = 5x+4 => x = color(green)(1/2)#

  • If #(3x+5)<0#, you have

#|3x+5| = -(3x+5) = -3x-5#

This will get you

#-3x-5 = 5x + 4 => x = color(red)(-9/8)#

Since #x=-9/8# does not satisfy the condition #x> -4/5#, this solution will be extraneous.

As a result, th only solution to this equation is #x=1/2#.