Rewrite the inequality as two inequalities that do not have absolute values. Replace the absolute value signs in one of the equations with a set of parentheses preceded by a plus sign. Do the same thing to get the second expression except use a negative sign. For example, the expression Y is less than |X - 3| - 4 can be replaced by the two expressions Y is less than +(X - 3} - 4 and Y is less than -(X - 3) - 4. This means that Y is less than X - 7 and Y is less than -X - 1.
Replace the inequality sign by an equal sign and graph to get the boundary of the inequality region. To find out which side of the line is the region, choose a point on one side and test it. If it fits the inequality, all the points on that side of the boundary will also fit the inequality. If it does not fit, then all the points that will fit are on the other side of the boundary. For example, if the boundary is the line Y = X - 7, look at the point (0, 0). It will not fit in the inequality Y is less than X - 7, so all the points that will fit are on the other side of the Y = X - 7 line.
Look for the overlapping regions -- this defines the region that contains the points where the original absolute value inequality is true. If you graph both Y = X - 7 and Y = -X - 1, and then test each one with the (0, 0 ) point you will find that the region that satisfies each equation is below the line. Therefore the region that satisfies both inequalities is the region below both lines. This region has a vertex at (3, -4). Notice that points in this region, like (3, -5) satisfy the original absolute value inequality Y is less than |X - 3| - 4.