Question

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

Answer 1

`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`.