Problem 9 - Algebra and Trigonometry

Question

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. $$ [-3, \infty) $$

Step-by-Step Solution

Answer

The set-builder notation for \([-3, \infty)\) is \(\{x | x \geq -3\}\). On the number line, -3 is marked by a closed circle, with an arrow pointing to the right, representing numbers up towards infinity.

Step 1: Translate interval notation to set-builder notation

In interval notation, \([-3, \infty)\) means all numbers greater than or equal to -3 and less than infinity. The corresponding set-builder notation is \(x | x \geq -3\). This means 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, and mark -3. Since the interval includes -3, shown by the square bracket in the interval notation, make a closed circle at -3 on the number line. Next, draw an arrow from -3 pointing to the right, which represents all numbers greater than -3 up to infinity.

Key Concepts

Set-builder Notation

Set-builder notation is a method of describing a set of numbers by specifying a property that its members must satisfy. It provides a precise way to describe mathematical sets without listing out every element, which becomes especially useful for infinite sets.
In the context of intervals, set-builder notation clarifies exactly what values are included in a set. For instance, when we have the interval \([-3, \infty)\) in interval notation, the equivalent in set-builder notation is \(x \mid x \geq -3\).
This expresses that the set contains all numbers \(x\) that are greater than or equal to \(-3\). Notice the use of the vertical bar, or "pipe" symbol \(|\), which acts as a shorthand for "such that."

• This notation helps to express complex sets simply by stating the rule they follow.
• Makes understanding properties of the set easier, especially when dealing with inequalities.

Number Line

The number line is a visualization tool crucial for understanding the placement and relationship of numbers. It is a horizontal line where each point corresponds to a number, and it helps in visualizing mathematical operations and relationships.
When graphing intervals, the number line makes it easier to see which numbers are included in or excluded from a particular interval.

• An arrow on one end typically signifies that numbers in that direction extend infinitely.
• Numbers increase as you move to the right and decrease as you move to the left.

For the interval \([-3, \infty)\), you would mark \(-3\) on this line and indicate the interval starts there and extends towards infinity to the right.

Graphing Intervals

Graphing intervals on a number line is an effective way to visually communicate which numbers are included in a set described by an interval.
Here's a step-by-step approach to graphing the interval \([-3, \infty)\):
1. Draw a horizontal line to represent the number line.2. Identify and mark the number \(-3\) on this line.
- Since the interval includes \(-3\), indicated by the square bracket, make a solid dot (or closed circle) at \(-3\).
3. Draw a line or arrow starting from the dot and extending to the right to showcase all numbers greater than \(-3\).
- This arrow represents the interval extending towards infinity, indicating there is no upper bound.

• Graphing intervals this way helps in visually confirming the range and limits of the set.
• Makes understanding inequalities related to the set intuitive.

Inequalities

Inequalities are mathematical expressions used to compare two values that are not necessarily equal. They express a range of possible solutions rather than specific values, such as \(x \geq -3\). These are represented on graphs and equations using symbols like \(>\), \(<\), \(\geq\), and \(\leq\).

• \(\geq\) means "greater than or equal to," indicating that the number on one side should be larger or equal to the other number.
• Inequalities communicate which part of a number line or coordinate plane falls within specified conditions.

For the interval \([-3, \infty)\), the inequality \(x \geq -3\) is used. This inequality means every number \(x\) on the number line that is \(-3\) or larger is a part of this set, extending up to infinity. By using inequalities, you can grasp the scope and boundaries of sets. It helps in easily modeling real-world scenarios where solutions aren't exactly one number but a range of possible values.