Problem 9 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$[-3, \infty)$$

Step-by-Step Solution

Answer

The interval \([-3, \infty)\) can be expressed in set-builder notation as \(x | x \geq -3\). The graph of this interval on a number line starts at -3 with a solid dot, extending to the right towards \(\infty\).

Step 1: Convert to Set-Builder Notation

Set-builder notation is a mathematical notation used to describe the properties of members of a set. For \([-3, \infty)\), this interval signifies all real numbers that are greater than or equal to -3. In set-builder notation, this would be expressed as: \(x | x \geq -3\). This is read as 'the set of all x such that x is greater than or equal to -3.'

Step 2: Graph the Interval on a Number Line

Draw a number line, labeling it with numbers greater and less than -3 for reference. Plot a solid dot on -3, denoting that -3 is included in the interval. Then, draw a line extending to the right side, following the direction of the larger numbers, to indicate that interval extends to \(\infty\). It is standard not to draw an end point for \(\infty\) as it is not a concrete number, but a concept that signifies unboundedness in the positive direction.