Rewrite each expression without absolute value bars: a. ∣1 − √2∣ b. ∣π − 3∣ c. ∣x∣/xSo this is a problem we have taken from the book Precalculus Essentials 5th Edition written by Robert F. Blitzer. Are you not able to find it in the book? Well do not worry! Just go to the Pre-requisites chapter P part of the book and look for problem Check Point 6. It is in page 10 of the book. So with that said, let us now try to solve this problem. But how do we do that?
The answer is to break down the problem into smaller parts. To do that, let us re-read the problem statement once again.
Rewrite each expression without absolute value bars:
So what is it asking us to do? It is asking us to find the value of each of the expressions, right? But there is a condition to be met. It wants us to find the value without using the absolute value bars.
So what does it mean?
It means that we need to get rid of the absolute value represented by the | | symbol. But how do we do that? To know more about it, let us first learn what an absolute value is. Alright?
What is an Absolute value?
When we try to solve any expression, we get an output value, right? But here is the thing. This output value can be either positive or negative. Still not clear? Well then, take a look at this example:
y = x + 3
Now, in the above expression, the value of y can be both positive or negative, right?
So if the value of x is 6, we get y = 9. But on the other hand, if x = -4, we get y = -1.
So as you can see from above, the final value of y can be positive or negative. It depends on the value of x.
Are you clear up to this point? Great!
So what do we do in case of an absolute value? Well if we want to find the absolute value of y, we will drop the sign symbol. So in that case, when x=-4, y=1!
So we represent absolute value of this equation using the formula:
y = | x + 3 |
So whenever you see the symbol | |, it means we are taking the absolute value of it where we drop the -ve sign.
In other words, you can also say that an absolute value is always +ve.
So now that we have learnt what an absolute value is, let us answer our actual question.
Rewrite each expression without absolute value bars
So let us start solving this question one by one. Let us take a look at the first problem:
a. ∣1 − √2∣
a. ∣1 − √2∣
So let us first break down the problem first. To begin with, we will drop the absolute value bars.
So the problem simply becomes: 1 − √2
But we know that the value of √2 = 1.4142
So what will be the value for (1 – 1.4142) ?
Yes, 1 – 1.4142 = -0.4142
So as you can see, we have a negative valued result. So the final answer will be:
Answer: 1 − √2 = -0.4142
In similar ways, we can now rewrite each expression without absolute value bars as shown below:
b. Rewrite each expression for ∣π − 3∣
So the expression here is
∣π − 3∣
which without absolute value becomes equal to:
π − 3
Now we know the value of π = 3.1415
So the resulting answer will be:
Answer: 3.1415 – 3 = 0.1415
So now coming to the final question, the answer is straight forward:
Answer: x/x = 1