Express an interval with an absolute value

I am looking at a problem in a review of basic set notation which reads as follows:

*(a) [1, 5]*

(b) (1, 4)

(c) [-1, 6)

(d) [-4, 4]

The interval in (d) may be expressed in the form {x | x is a number and |x| ≤ 4}. Two of the other intervals listed can similarly be expressed with the aid of absolute values. Find the two and display the result.

I assume the intervals you could do this with would be (a) and (b), but I don't really know where to go from there. Could anyone offer any hints in the right direction? Thanks!

Re: Express an interval with an absolute value

Quote:

Originally Posted by

**Ragnarok** I am looking at a problem in a review of basic set notation which reads as follows:

*(a) [1, 5]*

(b) (1, 4)

(c) [-1, 6)

(d) [-4, 4]

The interval in (d) may be expressed in the form {x | x is a number and |x| ≤ 4}. Two of the other intervals listed can similarly be expressed with the aid of absolute values. Find the two and display the result.

I assume the intervals you could do this with would be (a) and (b), but I don't really know where to go from there. Could anyone offer any hints in the right direction? Thanks!

The interval in (a) may be expressed in the form {x | x is a number and |x-3| ≤ 2}.

The interval in (b) may be expressed in the form {x | x is a number and |x-5/2| < 3/2}.

Re: Express an interval with an absolute value

In general

$\displaystyle c \leq x \leq d \equiv |x-a| \leq b \equiv a-b \leq x \leq a+b \implies a= \frac{c+d}{2} , b = \frac{d-c}{2} $

Regards,

Kalyan