How to solve the absolute-value inequalities?

Absolute value means how far away you are from zero. It's better to draw a number line to understand and solve the question. 

Given the inequality l 4x+3 l >15 , the distance of the 4x+3 value from 0 must be greater than 15, so 4x+3 has to be either greater than 15 or less than -15 (negative 15). so it becomes

4x+3 > 15 or 4x+3< -15

Then subtract 3 from both sides, 4x >12 or 4x < -18, 

divided by 4 , so the inequalities become x > 3 or x < -9/2 which are the solutions.