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

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Answer 1 · 2015-08-02T19:21:04

$x = 1/2$

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

$|3x + 5| = 3x+5$

and the equation becomes

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

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

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 $x=1/2$ .

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