Simplify An Expression Containing Absolute Values

In summary, the problem statement indicates that x is in the open interval (1, 2), meaning that x is greater than 1 and less than 2. This notation is used to describe a set of points that satisfy a certain condition, in this case, x > 1 and x < 2.
  • #1
nycmathguy
Homework Statement
Simplify an expression containing absolute values.
Relevant Equations
n/a
See attachment.

I don't understand the solution given by David Cohen. I am sure this is a shortcut explanation. I don't like shortcut explanations.

1. What in the problem indicates that x > 1?

2. What in the problem indicates that x < 2?
 

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  • #2
nycmathguy said:
1. What in the problem indicates that x > 1?
It's given information.
The problem statement says "given that x is in the open interval (1, 2)."
This means that x > 1 and x < 2.

nycmathguy said:
2. What in the problem indicates that x < 2?
This is also given information.
 
  • #3
Mark44 said:
It's given information.
The problem statement says "given that x is in the open interval (1, 2)."
This means that x > 1 and x < 2.This is also given information.
You said:

"It's given information.
The problem statement says "given that x is in the open interval (1, 2)."
This means that x > 1 and x < 2."

Sorry but I don't get it.
 
  • #4
nycmathguy said:
This means that x > 1 and x < 2."

Sorry but I don't get it.
What do you not get? Can you think of any numbers that are both a) larger than 1, and b) smaller than 2?
I can easily come up with 100 of them.
 
  • #5
nycmathguy said:
You said:

"It's given information.
The problem statement says "given that x is in the open interval (1, 2)."
This means that x > 1 and x < 2."

Sorry but I don't get it.
You know what an open interval is, right?

For me, I visualize an open interval by closing my eyes and thinking of a number line extending to the left and right. It is laid out with a zero in the middle. There are labels for -1, -2, etc extending to the left and labels for 1, 2, etc extending to the right.

On this line we lay a left parenthesis at the point labelled 1 and a right parenthesis at the point labelled 2. The open interval includes all of the points between the two parentheses. The word "open" means that the endpoints are not included in the interval.

Suppose that we pick an x value at random somewhere in the interval. We do not know what x is. But if it is in the interval, we know that it must be somewhere to the left of the 2. And we know that it must be somewhere to the right of the 1.

Which is the same as saying that "x > 1 and x < 2".
 
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  • #6
jbriggs444 said:
You know what an open interval is, right?

For me, I visualize an open interval by closing my eyes and thinking of a number line extending to the left and right. It is laid out with a zero in the middle. There are labels for -1, -2, etc extending to the left and labels for 1, 2, etc extending to the right.

On this line we lay a left parenthesis at the point labelled 1 and a right parenthesis at the point labelled 2. The open interval includes all of the points between the two parentheses. The word "open" means that the endpoints are not included in the interval.

Suppose that we pick an x value at random somewhere in the interval. We do not know what x is. But if it is in the interval, we know that it must be somewhere to the left of the 2. And we know that it must be somewhere to the right of the 1.

Which is the same as saying that "x > 1 and x < 2".
Now, I get.

(1, 2) means x > 1 and x < 2 because the number we select for x lies between 1 and 2. It does not lie before 1 and surely not after 2.
 
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  • #7
nycmathguy said:
(1, 2) means x > 1 and x < 2 because...

There's no reason for a "because". (1, 2) is notation for the set of points such that 1 < x < 2. You don't have to do any deduction.

This is called interval notation. Square brackets are commonly used for intervals that include the endpoints. If I say x is in [1, 2], that means x lies in the set of points such that ##1 \leq x \leq 2##.

We need some way to describe whether we're including the endpoint or not, so a parenthesis is commonly used by many people to indicate that the endpoint is not part of the interval.

If I say ##x \in [1, 2)##, note that I have a square bracket next to the 1, but a parenthesis next to the 2. That stands for the interval that includes 1 but does not include 2, ##1 \leq x \lt 2##

Some people use a backward square bracket for this notation. So I might write ]1, 2[ instead of (1, 2) to indicate the interval ##1 \lt x \lt 2##, and [1, 2[ to indicate the interval ##1 \leq x \lt 2##.

But either way, this is all notation. (1, 2) means that x > 1 and x < 2 because (1, 2) is what we call the set of points such that x > 1 and x < 2. No other meaning.

(No other meaning in this context. I know it looks like an ordered pair, but it is not an ordered pair. It's an interval. That's probably why some people prefer the ]1,2[ notation, to avoid possible confusion.)
 
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  • #8
nycmathguy said:
Now, I get.

It does not lie before 1 and surely not after 2.
That is an incomplete statement. It also does not lie before or ON 1 or after or ON 2
 
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  • #9
RPinPA said:
There's no reason for a "because". (1, 2) is notation for the set of points such that 1 < x < 2. You don't have to do any deduction.

This is called interval notation. Square brackets are commonly used for intervals that include the endpoints. If I say x is in [1, 2], that means x lies in the set of points such that ##1 \leq x \leq 2##.

We need some way to describe whether we're including the endpoint or not, so a parenthesis is commonly used by many people to indicate that the endpoint is not part of the interval.

If I say ##x \in [1, 2)##, note that I have a square bracket next to the 1, but a parenthesis next to the 2. That stands for the interval that includes 1 but does not include 2, ##1 \leq x \lt 2##

Some people use a backward square bracket for this notation. So I might write ]1, 2[ instead of (1, 2) to indicate the interval ##1 \lt x \lt 2##, and [1, 2[ to indicate the interval ##1 \leq x \lt 2##.

But either way, this is all notation. (1, 2) means that x > 1 and x < 2 because (1, 2) is what we call the set of points such that x > 1 and x < 2. No other meaning.

(No other meaning in this context. I know it looks like an ordered pair, but it is not an ordered pair. It's an interval. That's probably why some people prefer the ]1,2[ notation, to avoid possible confusion.)
Thanks. Good data here.
 
  • #10
Thank you everyone.
 

What is an absolute value?

An absolute value is the distance of a number from 0 on a number line. It is always a positive number.

How do I simplify an expression containing absolute values?

To simplify an expression with absolute values, you need to consider the different cases of the absolute value. If the expression inside the absolute value is positive, you can remove the absolute value signs. If the expression inside the absolute value is negative, you need to change the sign of the expression and then remove the absolute value signs.

Can I combine multiple absolute values in one expression?

Yes, you can combine multiple absolute values in one expression. However, you need to be careful when doing so and make sure to consider all the different cases of the absolute values.

What happens if the expression inside the absolute value is a variable?

If the expression inside the absolute value is a variable, you can still simplify the expression by considering the different cases of the absolute value. However, the final result may contain absolute values with the variable still inside.

Are there any specific rules for simplifying expressions with absolute values?

Yes, there are specific rules for simplifying expressions with absolute values. These rules involve considering the different cases of the absolute value and using algebraic techniques to simplify the expression. It is important to carefully follow these rules when simplifying expressions with absolute values.

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