How Do Absolute Values Express At Least and At Most Conditions?

[mathdad]

Rewrite each statement using absolute values.

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

Work:

| x - 4 | > or = 8

Correct?

Why must we write greater than or equal to for AT LEAST statements?

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

Work:

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

Correct?

Why must we write less than or equal to for AT MOST statements?

 

[MarkFL]

Rewrite each statement using absolute values.

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

Work:

| x - 4 | > or = 8

Correct?

Why must we write greater than or equal to for AT LEAST statements?

Yes, that's correct. When we say something is "at least" some value, that's equivalent to saying it is that value or greater. If I say I have at least \$20 in my pocket, then you know the money in my pocket is \$20 or more.

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

Work:

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

Correct?

Why must we write less than or equal to for AT MOST statements?

That's correct too. When we say some value is at most some other value, then that's equivalent to saying it is that value or less. If I say I have "at most" \$20 in my pocket then you know the money I have in my pocket is less than or equal to \$20. :D

 

[mathdad]

Good to know the difference between "at least" and "at most" because it is very common in the world of inequality applications.