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

[mathdad]

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?

 

[MarkFL]

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.

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:

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

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

| x - 1 | < 1/2

Correct. :D

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)

 

[mathdad]

Correct to me means I understood the textbook lecture.