Problem 47 - Precalculus Mathematics for Calculus

Question

Express the interval in terms of inequalities, and then graph the interval. $$[2,8)$$

Step-by-Step Solution

Answer

The interval \([2, 8)\) as inequalities is \(2 \leq x < 8\).

Step 1: Understand the Notation

The interval \([2, 8)\) is a mixed interval. The square bracket \([\ ]\) means that the number 2 is included in the interval. The parenthesis \(()\) means that the number 8 is not included in the interval. This indicates that the interval includes all numbers from 2 up to, but not including, 8.

Step 2: Write the Interval as an Inequality

Convert the interval notation \([2, 8)\) into an inequality. Since 2 is included in the interval, we write it as \(x \geq 2\). Because 8 is not included, we write \(x < 8\). Therefore, the interval in terms of inequalities is \(2 \leq x < 8\).

Step 3: Graph the Interval

To graph \([2, 8)\), start by drawing a number line. Mark the number 2 with a solid dot, indicating that it is included in the interval. Mark the number 8 with an open circle, showing it is not included. Shade the region between these two points on the number line to represent all numbers \(x\) such that \(2 \leq x < 8\).

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not exactly equal. They come in different forms, such as "less than" (<), "greater than" (>), "less than or equal to" (\(\leq\)), and "greater than or equal to" (\(\geq\)). These symbols help us describe ranges or intervals of values rather than single numbers.

When dealing with intervals, inequalities are used to express which values are included in a given set. For example, the inequality \(2 \leq x < 8\) describes all numbers \(x\) that are greater than or equal to 2 and less than 8. This inequality can also be thought of as a range between the numbers 2 and 8. The way this is expressed, using different inequality signs, helps clarify exactly which endpoint values (like 2 and 8 here) are included or excluded from the interval.

Graphing Intervals

Graphing intervals is a visual way of representing the numbers that satisfy given inequalities. By creating a graph, we can see at a glance which numbers are included in an interval.

To graph an interval like \([2, 8)\), we first draw a number line, which is a straight horizontal line representing a range of numbers. On this line, we mark the key points of the interval — in our case, the numbers 2 and 8. It is critical to indicate inclusion or exclusion of these boundary numbers:

• A solid dot is placed on a number that is included in the interval, such as 2 in \([2, 8)\).
• An open circle is used for a number that is not included, like 8 in our interval.

Once these points are marked, the interval is easily visualized by shading or highlighting the section of the number line between these points, showing that all numbers in this shaded section are part of the interval.

Number Line Representation

A number line is a simple yet powerful tool used in mathematics to visualize numeric values and operations involving them. It is a horizontal line marked with numbers in a sequential order, often with 0 at the center, going up as you move to the right, and going down as you move to the left.

In representing intervals on a number line, it helps clarify which numbers are part of the set we’re considering. Each number point on the number line has a specific representation style that gives additional information about the interval:

• Solid dots: Indicate that a number is included in the interval. For instance, in \([2, 8)\), the solid dot on 2 means 2 is part of the set.
• Open circles: Show that a number is not included in the interval. For instance, the 8 in \([2, 8)\) has an open circle.

The shading between these symbols further enhances understanding, signifying all the values between them are included in the interval. This approach not only makes it easy to communicate mathematical concepts but also aids in comprehension and problem-solving.