Sure, let's break down how you can express the solution to the inequality [tex]\(x > 1\)[/tex] using set-builder notation in a detailed, step-by-step manner.
1. Identify the Inequality:
The given inequality is [tex]\(x > 1\)[/tex], which tells us that we are looking for all values of [tex]\(x\)[/tex] that are greater than 1.
2. Understand Set-Builder Notation:
Set-builder notation is a way of specifying a set by stating the properties that its members must satisfy. It is usually written in the format:
[tex]\[
\{ \text{variable} \mid \text{condition} \}
\][/tex]
where the vertical bar [tex]\( \mid \)[/tex] can be read as "such that," and the condition is a characteristic that defines the elements of the set.
3. Construct the Set-Builder Notation:
For the inequality [tex]\(x > 1\)[/tex], we want to denote the set of all [tex]\(x\)[/tex] such that [tex]\(x\)[/tex] is greater than 1. In set-builder notation, this is written as:
[tex]\[
\{ x \mid x > 1 \}
\][/tex]
4. Interpret the Notation:
The set-builder notation [tex]\(\{ x \mid x > 1 \}\)[/tex] reads as "the set of all [tex]\(x\)[/tex] such that [tex]\(x\)[/tex] is greater than 1." This succinctly captures the solution to the inequality.
Combining all these steps, the solution to the given inequality [tex]\(x > 1\)[/tex] in set-builder notation is:
[tex]\[
\{ x \mid x > 1 \}
\][/tex]
