What is the solution set for $abs(4x – 3) – 2 > 3$ ?

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Answer 1 · 2018-03-23T11:09:56

$(-oo,-1/2)uu(2,oo)$

If we look at the definition of absolute value:

$|a|=a \ \ \ \ \ \ \ \$ if and only if $\ \ \ a>=0$

$|a|=-a \ \ \$ if and only if $\ \ \ a<0$

It follows from this, that we need to solve both:

$4x-3-2>3$ and $-(4x-3)-2>3$

$4x-3-2>3$

$4x-5>3$

$x>8/4$

$color(blue)(x>2)$

$-(4x-3)-2>3$

$-4x+3-2>3$

$-4x>2$

$color(blue)(x<-1/2)$

This gives us a union of intervals:

$(-oo,-1/2)uu(2,oo)$

What is the solution set for abs(4x – 3) – 2 > 3abs(4x – 3) – 2 > 3?