We pick up this post from where we left the post of last week in which we looked at a few properties of absolute values in two variables. There is one more property that we would like to talk about today. Thereafter, we will look at a question based on some of these properties.
(III) |x – y| = 0 implies x = y
x and y could be positive/negative integer/fraction; if the absolute value of their difference is 0, it means x = y. They cannot have opposite signs while having the same absolute value. They must be equal. This also means that if and only if x = y, the absolute value of their difference will be 0.
Mind you, this is different from ‘difference of their absolute values’
|x| – |y| = 0 implies that the absolute value of x is equal to the absolute value of y. So x and y could be equal or they could have opposite signs while having the same absolute value.
Let’s now take up the question we were talking about.
Question: Is |x + y| < |x| + |y|?
Statement 1: | x | ≠ | y |
Statement 2: | x – y | > | x + y |
Solution: One of the properties we discussed last week was
“For all real x and y, |x + y| <= |x| + |y|
|x + y| = |x| + |y| when (1) x and y have the same sign (2) at least one of x and y is 0.
|x + y| < |x| + |y| when (1) x and y have opposite signs”
We discussed in detail the reason absolute values behave this way.
So our question “Is |x + y| < |x| + |y|?” now becomes:
Question: Do x and y have opposite signs?
We do not care which one is greater – the one with the positive sign or the one with the negative sign. All we want to know is whether they have opposite signs (opposite sign also implies that neither one of x and y can be 0)? If we can answer this question definitively with a ‘Yes’ or a ‘No’, the statement will be sufficient to answer the question. Let’s go on to the statements now.
Statement 1: | x | ≠ | y |
This statement tells us that absolute value of x is not equal to absolute value of y. It doesn’t tell us anything about the signs of x and y and whether they are same or opposite. So this statement alone is not sufficient.
Statement 2:| x – y | > | x + y |
Let’s think along the same lines as last week – when will | x – y | be greater than | x + y |? When will the absolute value of subtraction of two numbers be greater than the absolute value of their addition? This will happen only when x and y have opposite signs. In that case, while subtracting, we would actually be adding the absolute values of the two and while adding, we would actually be subtracting the absolute values of the two. That is when the absolute value of the subtraction will be more than the absolute value of the addition.
For Example: x = 3, y = -2
| x – y | = |3 – (-2)| = 5
| x + y | = |3 – 2| = 1
x = -3, y = 2
| x – y | = |-3 – 2| = 5
| x + y | = |-3 + 2| = 1
If instead, x and y have the same sign, | x + y | will be greater than| x – y |.
If at least one of x and y is 0, | x + y | will be equal to| x – y |.
Since this statement tells us that | x – y | > | x + y |, it implies that x and y have opposite signs. So this statement alone is sufficient to answer the question with a ‘Yes’.
Takeaway from this question:
If x and y have the same signs, | x + y | >| x – y |.
If x and y have opposite signs, | x + y | <| x – y |.
If at least one of x and y is 0, | x + y | =| x – y |.
You don’t need to ‘learn this up’. Understand the logic here. You can easily recreate it in the exam if need be.
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!