Question

Express the fact that `x` differs from 2 by less than`1/ 2` as an inequality involving an absolute value. What is `x`?

Answer 1

`|x - 2| < frac(1)(2)`

`frac(3)(2) < x < frac(5)(2)`

Explanation:

What the statement is saying is that the magnitude of the difference between `x` and `2` will be less than `frac(1)(2)`.

The difference between `x` and `2` can be expressed as `x - 2`.

The magnitude of any value or expression can be found by using absolute value.

So the magnitude of the difference can be expressed as `|x - 2|`.

Now, this whole expression is less than `frac(1)(2)`, or `|x - 2| < frac(1)(2)`.

Let's solve this inequality using the properties of absolute value:

`Rightarrow x - 2 < frac(1)(2) Rightarrow x < frac(1)(2) + 2 therefore x < frac(5)(2)`

`or`

`Rightarrow x - 2 > - frac(1)(2) Rightarrow x > 2 - frac(1)(2) therefore x > frac(3)(2)`

If we combine these two results, we get the set `frac(3)(2) < x < frac(5)(2)`.

Therefore, `x` is a number between, but not including, `frac(3)(2)` and `frac(5)(2)`.