Is this the correct way to rewrite absolute value statements?

In summary, the statements can be rewritten using absolute values as | x - 4 | > or = 8 and | 4 - x | > or = 8 for the distance between x and 4 being at least 8. For the distance between x^3 and -1 being at most 0.001, it can be expressed as | x^3 -(-1) | < or = 0.001 or | - 1 - x^3 | < or = 0.001. The distance between x and 1 is less than 1/2 can be written as | x - 1 | < 1/2, and the distance between x and 1 exceeds 1/2
  • #1
mathdad
1,283
1
Rewrite each statement using absolute values.

1. The distance between x and 4 is at least 8.

| x - 4 | > or = 8

Can this also be expressed as | 4 - x | > or = 8?

If so, why?

2. The distance between x^3 and -1 is at most 0.001.

| x^3 -(-1) | < or = 0.001

Can this also be expressed as | - 1 - x^3 | < or = 0.001?

3. The distance between x and 1 is less than 1/2.

| x - 1 | < 1/2

4. The distance between x and 1 exceeds 1/2.

| x - 1 | > 1/2

Is any of this correct?
 
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  • #2
RTCNTC said:
Rewrite each statement using absolute values.

1. The distance between x and 4 is at least 8.

| x - 4 | > or = 8

Can this also be expressed as | 4 - x | > or = 8?

If so, why?

Your expressions are correct, and the reason you can express it either way is:

\(\displaystyle |-x|=|x|\)

The distance between 4 and x is the same as the distance between x and 4.

RTCNTC said:
2. The distance between x^3 and -1 is at most 0.001.

| x^3 -(-1) | < or = 0.001

Can this also be expressed as | - 1 - x^3 | < or = 0.001?

Yes, and you could even write:

\(\displaystyle \left|x^3+1\right|\le0.001\)

RTCNTC said:
3. The distance between x and 1 is less than 1/2.

| x - 1 | < 1/2

Correct. :D

RTCNTC said:
4. The distance between x and 1 exceeds 1/2.

| x - 1 | > 1/2

Is any of this correct?

It all looks good to me. (Yes)
 
  • #3
Correct to me means I understood the textbook lecture.
 

Related to Is this the correct way to rewrite absolute value statements?

What is an absolute value statement?

An absolute value statement is a mathematical expression that represents the distance of a number from zero on a number line. It is always positive, as it measures distance, not direction.

How do you solve an absolute value statement?

To solve an absolute value statement, you must isolate the absolute value expression on one side of the equation and then consider two cases: the expression inside the absolute value is either positive or negative. If it is positive, the solution is the value of the expression. If it is negative, the solution is the opposite of the value of the expression.

What happens when an absolute value statement is equal to zero?

If an absolute value statement is equal to zero, the value of the expression inside the absolute value must be zero. This means that the solution to the statement is the value of the expression itself.

Can absolute value statements have variables?

Yes, absolute value statements can have variables. When solving an absolute value statement with variables, the solution will be in terms of those variables.

What is the difference between an absolute value statement and an absolute value function?

An absolute value statement is an equation or inequality, while an absolute value function is a mathematical function. The absolute value function always returns a positive value, while an absolute value statement can have positive and negative solutions.

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