Problem 46 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$(-4,0]$$

Step-by-Step Solution

Answer

The inequality notation is \(-4 < x \leq 0\).

Step 1: Understand the Interval Notation

The interval \((-4, 0]\) expresses a range of numbers. The parenthesis \((-4\)) indicates that -4 is not included in this interval, while the bracket \(0]\) means that 0 is included. This tells us that the interval includes all numbers greater than -4 and less than or equal to 0.

Step 2: Convert to Inequality Notation

To convert the interval \((-4, 0]\) into inequality notation, we consider the fact that -4 is not part of the set (which is shown by using \(-4 < x\)), and 0 is part of the set (hence \(x \leq 0\)). Combining these, the inequality notation becomes:\[-4 < x \leq 0\]

Step 3: Sketch the Number Line

To show the interval on a number line, we draw a line and mark points corresponding to the numbers -4 and 0. We'll use an open circle at -4 to signify that this endpoint is not included and a closed circle at 0 to show that it is included. Shade the line segment between these two points to indicate all numbers between them are included in the interval.

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a range of numbers. In mathematics, when you see a notation like \((-4, 0]\), it represents a specific set of numbers. The parentheses \((-4\)) indicate that -4 is not included in the interval, while the square bracket \(0]\) shows that 0 is included.

Your interval \((-4,0]\) includes all numbers that are greater than -4 but less than or equal to 0. This notation is great because it simplifies complex sets into manageable and easily readable expressions. It's also visually helpful, making it easy to see at a glance which numbers are part of the set and which are not.

• Open interval: Both endpoints are not included, e.g. \((a, b)\).
• Closed interval: Both endpoints are included, e.g. \([a, b]\).
• Half-open or half-closed intervals: One endpoint is included, e.g. \((a, b]\) or \([a, b)\).

Number Line

A number line is a fantastic visual tool used in mathematics to represent numbers or intervals. When sketching a number line for the interval \((-4, 0]\), you draw a horizontal line and place markers at important points like -4 and 0. A number line helps visually demonstrate which numbers are included within a particular interval.

On a number line:

• Numberson the line show specific values, which can include integers, fractions, or decimals.
• Positions indicate size and order, with numbers increasing as you move to the right and decreasing as you move to the left.
• It helps show relationships between different numbers and makes it easy to locate intervals.

Using a number line makes it easier to understand the concept of inclusion and exclusion in intervals by using specific markers such as open and closed circles.

Open Circle

An open circle is used on a number line to indicate that a particular number is not included in an interval. For example, when graphing the interval \((-4, 0]\), an open circle is drawn at -4. This clearly shows that -4 is not counted as part of the interval.

Open circles are essential because:

• They visually convey that the endpoint is not part of the set.
• They help in distinguishing between inclusive and exclusive values.
• They enhance understanding of interval notation by partnering with closed circles to show complex sets.

The open circle makes it very clear where the interval starts without including the boundary point, helping students quickly grasp where the range begins.

Closed Circle

A closed circle on a number line signifies that the number it touches is part of the given interval. In the interval \((-4, 0]\), a closed circle appears at 0 to show that 0 is included. This notation makes it clear and easy to see which endpoints are part of the set and which are not.

Here are some points about closed circles:

• They indicate inclusivity of the endpoint in the interval.
• They provide contrast with open circles, highlighting differences in sets.
• On a number line, they make understanding graphs and solutions simpler and more explicit.

Having a closed circle at 0 assures that 0 is part of the interval solution, making comprehension straightforward and ensuring you don't accidentally overlook included values.