Question

# Express the interval in terms of an inequality involving absolute value. (0,4)

Piecewise-Defined Functions
Express the interval in terms of an inequality involving absolute value.
(0,4)

2021-02-23

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
$$\displaystyle{c}={\frac{{{a}+{b}}}{{{2}}}}$$
$$\displaystyle{c}={\frac{{{0}+{4}}}{{{2}}}}$$
$$\displaystyle{c}={\frac{{{4}}}{{{2}}}}$$
$$\displaystyle{c}={2}$$
$$\displaystyle{r}={\frac{{{b}-{a}}}{{{2}}}}$$
$$\displaystyle{r}={\frac{{{4}-{0}}}{{{2}}}}$$
$$\displaystyle{r}={\frac{{{4}}}{{{2}}}}$$
$$\displaystyle{r}={2}$$
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 $$\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{2}{\mid}{<}{2}\rbrace$$