Step 1

To express the given interval in terms of an inequality involving absolute value.

Step 2

Given information:

The interval is (0, 4).

Step 3

Calculation:

Let (a, b) is an open interval and r is the radius, and the midpoint is c, then the interval can be represented as

\(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{c}{\mid}{<}{r}\rbrace\)

where \(\displaystyle{c}={\frac{{{a}+{b}}}{{{2}}}}\) and \(\displaystyle{r}={\frac{{{b}-{a}}}{{{2}}}}\)

And the given interval is (0, 4).

So, midpoint is

\(\displaystyle{c}={\frac{{{a}+{b}}}{{{2}}}}\)

\(\displaystyle{c}={\frac{{{0}+{4}}}{{{2}}}}\)

\(\displaystyle{c}={\frac{{{4}}}{{{2}}}}\)

\(\displaystyle{c}={2}\)

And radius is

\(\displaystyle{r}={\frac{{{b}-{a}}}{{{2}}}}\)

\(\displaystyle{r}={\frac{{{4}-{0}}}{{{2}}}}\)

\(\displaystyle{r}={\frac{{{4}}}{{{2}}}}\)

\(\displaystyle{r}={2}\)

Step 4

So, the inequality involving absolute value will be

\(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{c}{\mid}{<}{r}\rbrace\)

\(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{2}{\mid}{<}{2}\rbrace\)\ \ [substitute\ values\ of\ r\ and\ c]

Thus, the inequality is \(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{2}{\mid}{<}{2}\rbrace\)