Problem 45 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$[0,3)$$

Step-by-Step Solution

Answer

The interval \([0,3)\) is expressed as the inequality \(0 \leq x < 3\).

Step 1: Understand the Interval Notation

The interval \([0,3)\) includes all real numbers from 0 to just below 3. The square bracket \([\ ]\) means 0 is included in the interval, while the parenthesis \((\ )\) means 3 is not included.

Step 2: Express Using Inequality Notation

To express this interval in inequality notation, we look at the interval bounds and inclusion properties. Since 0 is included and 3 is not, this can be expressed as \(0 \leq x < 3\).

Step 3: Draw on a Number Line

Draw a horizontal line to represent the number line. Place markings for 0 and 3. Represent inclusion of 0 by using a filled dot at 0, and represent the exclusion of 3 by using an open dot for 3. Shade the region between and including 0 up to but not including 3.

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand used to describe a set of numbers within a specified range. For the interval \([0,3)\), this means it's talking about all real numbers starting from 0 and going up to, but not reaching 3. The square bracket "\([\ ]\)" at 0 means the interval includes the number 0 itself. On the other hand, the parenthesis "\((\ )\)" at 3 indicates that the number 3 is not included within this interval.

Using interval notation simplifies the representation of solutions to inequalities and the sets of real numbers they describe. It quickly displays whether the boundary numbers are part of the set or not, allowing for easy understanding when solving equations and graphically representing intervals.

No matter the numbers involved, interval notation consistently follows the rule where "\([\ ]\)" means 'included' and "\((\ )\)" means 'not included'. This makes it quite convenient and widely used in mathematics to express ranges of values concisely.

Number Line

The number line provides a visual way of representing numbers and understanding intervals. Imagine a horizontal line that stretches infinitely in both directions, with each point representing a real number. To illustrate the interval \([0,3)\) as mentioned in the exercise, you begin by finding and marking the numbers 0 and 3 on this line.

The number 0 will have a filled dot above it, showing it's included in the set, corresponding to the square bracket in the interval notation. The number 3, however, will have an open dot placed above it, indicating it's not part of the set, aligning with the bracket style. Between these dots, the space is shaded or highlighted to mark the numbers that fall within the interval.

This simple graphical representation on a number line not only clarifies what the interval includes but also aids in understanding related concepts, such as continuity or gaps between numbers. Number lines are practical tools for anyone trying to grasp the size and coverage of different number sets.

Real Numbers

Real numbers are a broad category covering all possible values that can exist on the number line, including both rational and irrational numbers. They encompass integers, fractions, and roots, among others. In the context of expressing intervals like \([0,3)\), understanding real numbers means recognizing that any value between the start and end points can be part of the interval, provided it adheres to the inclusion or exclusion conditions.

Real numbers are significant when dealing with inequalities because intervals like \([0,3)\) specify the range of real numbers a variable may satisfy. These numbers stretch out to infinity in both directions on a number line, allowing mathematicians to define the span of possible solutions for equations without bounds.

Each interval notation directly communicates a segment of real numbers, proving the importance of comprehending this concept in mathematical graphs and relations. So, when you see an interval, identify it not just as a static range but as a dynamic set of real numbers meeting specific criteria.