Absolute Values

Let’s look at an absolute value concept involving two variables. It is unlikely that you will see such a question on the actual, since it involves multiple steps, but it will help you understand absolute values better. Recall the definition of absolute value: |x| = x if x ≥ 0 |x| = -x if x < 0 So, to remove the absolute value sign, you will need to consider two cases – one when x is positive or 0, and another when it is negative. Say, you are given an inequality, such as |x – y| < |x|. Here, you have two absolute value expressions: |x – y| and |x|. You need to get rid of the absolute value signs, but how will you do that? You know that to remove the absolute value sign, you need to consider the two cases. Therefore: |x – y| = (x – y) if (x – y) ≥ 0 |x – y| = – (x – y) if (x – y) < 0 But don’t forget, we also need to remove the absolute value sign that |x| has. Therefore: |x| = x if x ≥ 0 |x| = -x if x < 0 In all we will get four cases to consider: Case 1: (x – y) ≥ 0 and x ≥ 0 Case 2: (x – y) < 0 and x ≥ 0 Case 3: (x – y) ≥ 0 and x < 0 Case 4: (x – y) < 0 and x < 0 Let’s look at each case separately: Case 1: (x – y) ≥ 0 (which implies x ≥ y) and x ≥ 0 |x – y| < |x| (x – y) < x -y < 0 Multiply by -1 to get: y > 0 In this case, we will get 0 < y ≤ x. Case 2: (x – y) < 0 (which implies x < y) and x ≥ 0 |x – y| < |x| -(x – y) < x 2x > y x > y/2 In this case, we will get 0 < y/2 < x < y. Case 3: (x – y) ≥ 0 (which implies x ≥ y) and x < 0 |x – y| < |x| (x – y) < -x 2x < y x < y/2 In this case, we will get y ≤ x < y/2 < 0. Case 4: (x – y) < 0 (which implies x < y) and x < 0 |x – y| < |x| -(x – y) < -x -x + y < -x y < 0 In this case, we will get x < y < 0. Considering all four cases, we get that both x and y are either positive or both are negative. Case 1 and Case 2 imply that if both x and y are positive, then x > y/2, and Case 3 and Case 4 imply that if both x and y are negative, then x < y/2. With these in mind, there is a range of values in which the inequality will hold. Both x and y should have the same sign – if they are both positive, x > y/2, and if they are both negative, x < y/2. Here are some examples of values for which the inequality will hold: x = 4, y = 5 x = 8, y = 2 x = -2, y = -1 x = -5, y = -6 etc. Here are some examples of values for which the inequality will not hold: x = 4, y = -5 (x and y have opposite signs) x = 5, y = 15 (x is not greater than y/2) x = -5, y = 9 (x and y have opposite signs) x = -6, y = -14 (x is not less than y/2) etc. As said before, don’t worry about going through this method during the actual  exam – if you do get a similar question, some strategies such as plugging in values and/or using answer choices to your advantage will work. Overall, this example hopefully helped you understand absolute values a little better.