Modulus Function | x | - Kamal Lohia
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We know from our school times that |x| is always positive, but in later days of our student life we encounter a more refined definition as,
So it is different than earlier ones? Certainly not.
But still doubts many and many a times that how is it possible |x| = -x. (i.e. negative of x)
Just be cautious. |x| is always positive. But for negative x we need to multiply it with a negative sign to make it positive.
One more inference: How many different values can be obtained by x/|x|?
Certainly it can be 1 or -1 only for non-zero x as the given expression (known as Signum Function, more commonly) is not defined for x = 0.
Now look at some other type of problems:
If x + y = 10, then what is the minimum value of |x| + |y|?
If x + y = -10, then find the maximum value of |x| - |y|?
If |x| + |y| = 10, then find the minimum value of (x - y)?
If |x| - |y| = 10, then find the maximum value of (x + y)?
I am leaving these as practice exercise for you.
|x - a| as ‘Distance Function’
What does ‘2’ means on a number line?
You may find the question absurd. But it is not.
See carefully that on number line a point’s name tells about it distance from Origin. Do you agree?
Let’s take some examples: ‘2’ is at a distance of 2 units from origin.
‘111’ is at a distance of 111 units from origin.
What about ‘-5’?
No problem ‘-5’ also tells that it is at a distance of 5 units from origin but on left side.
Similarly, what does ‘x’ talk about?
Same, distance of x from origin.
One more query now: What does x – 2 represent on number line?
Certainly going by the same logic, it tells about the difference of distances of x and 2 from the origin. Or in other words, I can also says that (x - 2) tells about the distance between x and 2.
A positive value will indicate that x lies on the right of 2 and negative indicates that x lies on the left of 2.
I hope it is clear so far.
Now comes the final query: What does |x - 2| represent on number line?
Certainly it represents the magnitude of distance between two points ‘x’ and ‘2’ on number line.
Now you must be able to see this question with a different vision and answer it easily.
Find x such that |x - 2| = 5
It simply means distance between ‘x’ and ‘2’ on number line is 5. That means x lies at a distance of 5 units from ‘2’ on either side i.e. x = 5 ahead of 2 i.e. 7 or 5 before 2 i.e. -3.
One more to clarify the concept: Find the minimum value of |x - 2| + |x - 3|.
Now it’ll clear this all properly.
Given expression represents sum of distances between ‘x’ and ‘2’ AND ‘x’ and ‘3’. Or it is sum of distances of x from 2 and 3.
Certainly, for the least case, x should lie between 2 and 3. And the least sum is distance between ‘2’ and ‘3’ i.e. 1.
Some equations involving two variables
Find the number of integral solutions of the equation |x| + |y| = 10.
You can simply make four equations: x + y = 10; x – y = 10; -x + y = 10; -x – y = 10 and plot them on co-ordinate plane to form a square.
Now required number of points can easily be counted as 11 on each line but vertex points have been counted twice. So final answer will be 4(11) – 4 = 4(10) = 40.
What about the area of the region bounded by the curve |x| + |y| = 10?
Clearly it is area of square with diagonal length of 20 i.e. ½ × 202 = 200.
One more variation:
Find the number of points with integral co-ordinates which satisfy |x| + |y| < = 10.
Not necessary to apply much brain. :)
Although there are different methods to solve this one but easiest to count the cases for |x| + |y| = 10 and |x| + |y| = 9 and |x| + |y| = 8 and so on up to |x| + |y| = 0 which comes out to be 4(10 + 9 + 8 + ... + 1) + 1 = 221.
What about the number of integral points which satisfy |x - 2| + |y + 3| = 10?