Question

express the interval in set - builder notation and graph the interval on a number line.
$(-infty, - 2)$
select the correct choice below and, if necessary, fill out the answer box to complete your choice.
a. the solution set in set - builder notation is ${xmidsquare}$.
(type an inequality or a compound inequality.)
b. the solution set is all real numbers.
c. there is no solution.

Explanation:

Step1: Understand Interval Notation

The interval \((-\infty, -2)\) means all real numbers \(x\) such that \(x\) is less than \(-2\). In interval notation, a parenthesis means the endpoint is not included. So for \((-\infty, -2)\), \(x\) satisfies \(x < -2\).

Step2: Set - Builder Notation

Set - builder notation is of the form \(\{x\mid \text{condition on }x\}\). For the interval \((-\infty, -2)\), the condition is \(x < -2\). So the set - builder notation is \(\{x\mid x < -2\}\). And since the interval is not all real numbers (option B is wrong) and there are solutions (option C is wrong), we choose option A.

Answer:

A. The solution set in set - builder notation is \(\{x\mid x < - 2\}\)

For graphing the interval \((-\infty, -2)\) on a number line:

1. Draw a number line.
2. Locate the point \(-2\) on the number line.
3. Since \(-2\) is not included in the interval, we draw an open circle at \(-2\).
4. Then, we draw an arrow starting from the open circle at \(-2\) and going to the left (towards negative infinity) to represent all numbers less than \(-2\).