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

Question
Piecewise-Defined Functions
asked 2021-02-22
Express the interval in terms of an inequality involving absolute value.
(0,4)

Answers (1)

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
\(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{c}{\mid}{<}{r}\rbrace\)</span>
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
\(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{c}{\mid}{<}{r}\rbrace\)</span>
\(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{2}{\mid}{<}{2}\rbrace\)</span>\ \ [substitute\ values\ of\ r\ and\ c]
Thus, the inequality is \(\displaystyle{\left\lbrace{x}\in{R}\right|}{x}-{2}{\mid}{<}{2}\rbrace\)</span>
0

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