Problem 45 - Precalculus Mathematics for Calculus

Question

45-50 - Express the interval in terms of inequalities, and then graph the interval. $$(-3,0)$$

Step-by-Step Solution

Answer

The interval (-3, 0) is represented by \\(-3 < x < 0\\) and is graphed with open circles on \\(-3\\) and \\(0\\), connecting them with a line.

Step 1: Identify the Interval Type

The given interval is \((-3, 0)\). Since it uses parentheses, it is an open interval, meaning the endpoints are not included.

Step 2: Convert Interval to Inequality

For the interval \((-3, 0)\), the inequality representation is \-3 < x < 0\. This means that \(-3\) is less than \(x\), and \(x\) is less than \(0\), but \(x\) cannot be exactly \(-3\) or \(0\).

Step 3: Prepare to Graph the Interval

To graph the interval on a number line, we need to show numbers between \(-3\) and \(0\), excluding the endpoints.

Step 4: Draw the Number Line

Draw a horizontal line and mark points representing \(-3\) and \(0\). These points will help visualize the interval.

Step 5: Plotting the Open Interval

On the number line, place an open circle at \(-3\) and another open circle at \(0\) to indicate that these numbers are not included. Then, draw a line connecting these open circles to show all numbers between \(-3\) and \(0\) are included.

Key Concepts

Understanding Interval Notation

Interval notation is a way of writing subsets of the real number line. It is a concise method to denote intervals, typically describing a range of numbers. In interval notation, we use parentheses \( ( \) and \( ) \) for open intervals and brackets \( [ \) and \( ] \) for closed intervals.
An open interval like \((-3, 0)\) implies that the endpoints -3 and 0 are not part of the interval. In contrast, a closed interval \[a, b\] includes its endpoints a and b.
Mixing these symbols allows for half-open intervals, where one endpoint is included, and the other is not. For example, \([-3, 0)\) indicates that -3 is included, but 0 is not. Choosing the right type of interval notation is essential to clearly convey which numbers are part of the set.

Exploring Inequalities

Inequalities express the relation between two expressions by using signs like <, >, \(\leq\), or \(\geq\). They allow us to compare values and represent ranges.
When we convert \((-3, 0)\) from interval notation to an inequality, it becomes \(-3 < x < 0\), conveying that x can be any number greater than -3 and less than 0.

• "<" and ">" are used for open intervals, just like parentheses.
• "\(\leq\)" and "\(\geq\)" correspond to closed intervals, using brackets.

If you encounter multiple inequality constraints, combining them can provide more specific conditions of a solution set. Practicing with inequalities helps develop a better understanding of how numbers relate and how to manipulate these comparisons effectively.

Graphing Intervals on a Number Line

Graphing intervals is a visual way to represent subsets of numbers. It's particularly useful when solving equations or inequalities as it provides an immediate way to show which values are part of a solution.
To graph the interval \((-3, 0)\), you start by sketching a simple horizontal line and marking points for -3 and 0.

• Open circles are used to mark that the endpoints -3 and 0 are not included in the interval.
• A solid line is drawn between the open circles to signify all numbers in between are included.

This graph serves as a quick reference and helps in understanding the scope of possible solutions in relation to the problem. Knowing how to translate between interval notation, inequalities, and a graph enables you to approach math problems with a fuller toolkit, making complex concepts simpler.