# Thread: Distance On A Number Line

1. ## Distance On A Number Line

The distance between a and n is |a - b| = |b - a|.

Using the definition given above, rewrite the statement using absolute value.

The distance between x and 4 is at least 8.

The distance between x and 4 is | x - 4|.

The |x - 4| is at least 8 can be expressed as

| x - 4| is > or = to 8.

Question:

Why must we set |x - 4| to be greater than or equal to 8 and not less than or equal to 8?

2. ## Re: Distance On A Number Line

Originally Posted by mathdad1965
Why must we set |x - 4| to be greater than or equal to 8 and not less than or equal to 8?
because they asked for the set of points whose distance from 4 is at least 8

$|x-4| \leq 8$ is the set of points whose distance from 4 is at most 8

3. ## Re: Distance On A Number Line

Originally Posted by romsek
because they asked for the set of points whose distance from 4 is at least 8

$|x-4| \leq 8$ is the set of points whose distance from 4 is at most 8
I need a clearer understanding.

1. Why is less than or equal associated with AT MOST?

2. Why is greater than or equal to associated with AT LEAST?

4. ## Re: Distance On A Number Line

Originally Posted by mathdad1965
I need a clearer understanding.

1. Why is less than or equal associated with AT MOST?

2. Why is greater than or equal to associated with AT LEAST?
I would think it moderately difficult to get along in life if you don't understand what at least and at most mean.

Suppose you wanted to spend some money on your hobby and your wife said ok but you can spend at most \$40 on it. Would you go spend \$50? No. You'd spend \$0 - \$40.

Suppose your wife's birthday is coming up and she warned you you had better spend at least \$50 on her gift. Would you spend \$10? No. You'd spend \$50 or more.$x \text { is at least } A \Rightarrow x \geq Ax \text { is at most } A \Rightarrow x \leq A$5. ## Re: Distance On A Number Line Originally Posted by romsek I would think it moderately difficult to get along in life if you don't understand what at least and at most mean. Suppose you wanted to spend some money on your hobby and your wife said ok but you can spend at most \$40 on it.

Would you go spend \$50? No. You'd spend \$0 - \$40. Suppose your wife's birthday is coming up and she warned you you had better spend at least \$50 on her gift.

Would you spend \$10? No. You'd spend \$50 or more.

$x \text { is at least } A \Rightarrow x \geq A$

$x \text { is at most } A \Rightarrow x \leq A$
Your explanation hit the nail on the head, so to speak. I totally get it now. I am going to use the "hobby" and "wife's birthday" examples to explain the concept of AT LEAST AND AT MOST to others seeking help in this regard. Great job!