1. Absolute Values

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^3 and -1 is at most 0.001.

The distance between x^3 and -1 is | x^3 + 1|.

The |x^3 + 1| is at most 0.001 can be expressed as

|x^3 + 1| is < or = 0.001.

Question:

Why must we set |x^3 + 1| to be less than or equal to 0.001and not greater than or equal to 0.001?

2. Re: Absolute Values

review the definitions of "at least" and "at most"

3. Re: Absolute Values

Why must we set |x^3 + 1| to be less than or equal to 0.001and not greater than or equal to 0.001?
The statement $|x^3+1|\ge 0.001$ means that the distance from $x^3$ to $-1$ is at least $0.001$.

4. Re: Absolute Values

Correction:

In the definition given, n should be b.

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

5. Re: Absolute Values

Originally Posted by romsek
review the definitions of "at least" and "at most"
The textbook is not too clear on the AT LEAST AND AT MOST definitions.

1. Can you provide a simple explanation?

2. In what way is the concept of AT LEAST AND AT MOST related to the number line?

6. Re: Absolute Values

The textbook is not too clear on the AT LEAST AND AT MOST definitions.

1. Can you provide a simple explanation?

2. In what way is the concept of AT LEAST AND AT MOST related to the number line?
$x \text{ is at least } a \text { if } a \leq x$

$x \text { is at most } a \text{ if } x \leq a$

with regard to the number line.

$\text{ if } x \text{ is at least } a \text{, then } x \in [a , \infty)$

$\text{ if } x \text{ is at most } a \text{, then } x \in (-\infty, a]$

7. Re: Absolute Values

Originally Posted by romsek
$x \text{ is at least } a \text { if } a \leq x$

$x \text { is at most } a \text{ if } x \leq a$

with regard to the number line.

$\text{ if } x \text{ is at least } a \text{, then } x \in [a , \infty)$

$\text{ if } x \text{ is at most } a \text{, then } x \in (-\infty, a]$
x is at least a if a is less than or equal to x.
Shouldn't it be if a is greater than or equal to x?

8. Re: Absolute Values

x is at least a if a is less than or equal to x.
Shouldn't it be if a is greater than or equal to x?
no

x is at least a means $x \geq a$

x is a or larger

9. Re: Absolute Values

Originally Posted by romsek
no

x is at least a means $x \geq a$

x is a or larger
Your inequality symbol is correct now but not in the previous reply. See it?

10. Re: Absolute Values

Note: x is at least a means that x > or = to a not x < or = to a. Correct?