What is the meaning when we write |x| > 1
Answers
Step-by-step explanation:
It means that x can be any positive number over 1 or any negative number under -1.
In math notation the answer is:
x < -1 or x > 1.
Explanation:
|x| is the absolute value of a number.
When x is negative, say -9 or -53 then the absolute numbers are 9 and 53.
So |-9| = 9
and |9| = 9
Also |-53| = 53.
and |53| = 53.
In math notation the definition of |x| is:
|n| = n if n >= 0.
|n| = -n if n < 0
In a more intuitive way for any n (which is zero or a positive number),
the absolute value is the positive distance from zero:
If n >= 0
|n| = n
|-n| = n
So the answer to the question is:
|x| > 1 means
any x where x < -1 or x > 1
Another way of putting it is
any x, where x is not between -1 to 1 inclusively. (Meaning not -1 and not 1)
Further explanation: Lets see what happens to |x| it is less than 1 and if the result is larger than 1.
We’ll start with positive numbers.
Lets start with a very small number less than 1 but above 0.
Lets try 0.1. The absolute value of 0.1 is 0.1. Not bigger than one.
Lets try 0.5. The absolute value of 0.5 is 0.5. Not bigger than one.
Lets try 0.9. The absolute value of 0.9 is 0.9. Not bigger than one.
Lets try 1. The absolute value of 1 is 1. Not bigger than 1.
So any positive n that is less or equal to 1 is NOT an answer to |n| > 1.
Well what poitive n DOES have an absolute value more than 1?
Simple: positive numbers that are larger than 1, will have an absolute value larger than 1.
|100| = 100
|724,536| = 724,536
Of course these results are larger than 1.
So for positive numbers
where x>1 then |x| > 1.
We now know that |x| > 1 is true for all x>1.
And untrue for 0 < x <= 1
What about negative numbers?
We’ll try a small fraction between zero and -1.
Lets try -0.1 The absolute value of -0.1 is 0.1. Not larger than one.
Lets try -0.5. The absolute value of -0.5 is 0.5. Not bigger than one.
Lets try -0.9. The absolute value of -0.9 is 0.9. Not bigger than one.
Lets try -1. The absolute value of -1 is 1. Not bigger than 1.
Well the absolute of what negative n DOES have an absolute value of more than 1?
Simple. Any number under -1 including -1.001 and -2 and -3000 all have an absolute value (1.001, 2, 3000) which is larger than
x = 1. |x| = 1.
First we’ll try x=-1000 Here goes: |-1000| = 1000. Yes definitely larger than 1.
For x=-100. We try |-100| which results in 100. Also larger than 1.
How about x=-724,536. Well |-724,536| = 724,536. Definitely larger than 1.
What about close to -1 but further from zero than -1.
Let’s try |-1.001| = 1.001 YES!
So any number x which is lower than -1 will give an absolute value larger than 1.
To sum it up:
When you say |x| > 1 it means:
for positive x’s: any x > 1
for negative x: any x < -1.
Which is summarized as the answer:
x < -1 or x > 1.