How does this make sense?

Question

In my books example it says:

whenever 0 < |x| < δ

Now , x=δ/2 and x=-δ/2 satisfy the above inequality.

How does x=-δ/2 satisfy the above inequality because -δ/2 is clearly less than zero and the inequality clearly says that abs value of x is b/w 0 and δ not less than zero.

Answer 1

I'm just going to repeat something you said: "the inequality clearly says that abs value of x is b/w 0 and δ not less than zero." That's correct--what we need to clarify is the definition of absolute value.

The absolute value is the DISTANCE from zero. So if you picture a number line, "2" is 2 units from zero. "-2" is ALSO two UNITS from zero. The only difference is -2 goes in the "negative" direction while "2" goes in the "positive" direction. But absolute value is not concerned with what the number actually is; it's only concerned with how far it is from zero. The technical term for this is magnitude.

Let's take another example. Let's define forward movement as the positive direction and backward movement as the negative direction. If I walk 10 feet forward, I've moved 10 feet, or +10 feet. If I walk 10 feet backward, I've still walked 10 feet, but this time I'm going to describe it as -10 feet. The sign gives us a little more information about what happened, but the distance I moved is the same in both cases: 10 feet. In other words, the magnitude of my distance traveled is always 10 feet. That's what absolute value is getting at.

Another way to describe absolute value is just to say if you have a positive number you keep it positive, but if you have a negative number you make it positive.

Let's look at it using math symbols:

If x = 10

Then |x| = 10

If x = -10

Then |x| = 10

So for your book's example:

x = -δ/2

|x| = δ/2

If the absolute value bars weren't there, you'd be correct by saying -δ/2 is NOT 0<x<δ. But because the absolute value bars are there you have to change the "-" to a "+".

All this problem is trying to say is that if you halve the distance to zero on the positive side of the number line, you also halve the distance to zero on the negative side.

Best of luck, hope this helps!

Answer 2

When x is -δ /2 the absolute value of x is + δ/2 which < δ