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

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The discussion confirms that the absolute value expressions for the given statements are correct. The distance between two points can be expressed in either order, as shown by the property that |x| = |-x|. The participants validate the expressions for distances involving x and constants, including x^3 and -1. They also agree that all rewritten statements accurately represent the original conditions. Overall, the rewrites are deemed correct and consistent with mathematical principles.
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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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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:

$$|-x|=|x|$$

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:

$$\left|x^3+1\right|\le0.001$$

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)
 
Correct to me means I understood the textbook lecture.
 

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