The section of absolute value inequalities is always difficult for students to understand. The first question students ask is, "what is the process to solve for the inequality for x?" I tell my students first to draw the picture of the inequality on a number line. A lot of students who are introduced to the concept for the first time cannot draw the inequality on the number line.
This is how I introduce the absolute value inequalities to my students. The absolute value of x (|x|) is the distance between some point with coordinate x on the number line to zero. For the absolute value inequality iii) in the picture given, |x - k| represents the distance from a point with coordinate x on the number line to some point with coordinate k. In the given picture, you can clearly see that the distance |x - k| is greater than c. Therefore the solution set must be x < k - c or x > k + c.
Just by drawing the definition of the absolute value on the number line, a student will be able to arrive at an answer without any algebra involved. After showing my students how to draw the absolute on the number line, I then show them how to set up the problem algebraically and then solve for the inequality. The conventional way students usually see this is by someone telling them to take the inside part of the absolute value, x - k, and make it less than - c or greater than c.
x - k < -c or x - k > c
Therefore, x < k - c or x > k + c.
By teaching the students the concepts by having them draw a picture of the number line first, the students will remember how to derive the solution set instead of simply following a set of rules with absolute value inequalities.