Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Plot the two real numbers on the real number line and place the appropriate inequality symbol \((<\) or \(>)\) between them.\(\frac{16}{3}, 1\)

Step-by-Step Solution

Answer

The inequality symbol between \(\frac{16}{3}\) and \(1\) is \(\frac{16}{3}>1\), as plotted on a number line.

Step 1: Convert to Decimal Form

Firstly, convert the given fraction \(\frac{16}{3}\) to a decimal. This should be straightforward division to achieve the decimal form.

Step 2: Compare Numbers

The decimal equivalent of \(\frac{16}{3}\) is approximately \(5.33\). Now, compare this number with \(1\). Clearly, \(5.33\) is greater than \(1\), so the inequality \(>\) will be used.

Step 3: Plot on Number Line

Place the two numbers \(1\) and \(5.33\) on the number line. Because \(5.33\) is greater than \(1\), place it to the right side of \(1\) on the number line. The inequality symbol \(>\) should point towards \(1\), and the wide part should face \(5.33\).

Key Concepts

Real Numbers

Real numbers are the extended family of numbers we deal with in everyday life. This includes integers, which are whole numbers without any fractional or decimal part, like \(1\) or \(-3)\). Additionally, it contains all the fractions and decimals you can imagine—numbers like \(\frac{1}{2}\), \(5.33\), and \(-2.75)\). Real numbers also encompass irrational numbers such as \(\pi\) and the square root of \(2\), which cannot be expressed as simple fractions.
When working with real numbers, you will often need to compare and plot them. Knowing the properties of real numbers can help you understand their behavior in equations and inequalities. The set of real numbers is essentially the entire number line, stretching infinitely in both directions.

Number Line

The number line is a visual tool that helps us understand the position and order of numbers. Imagine it as a long, straight line marked with zero in the center. To the right, the numbers increase positively, while to the left, they decrease negatively.
Plotting numbers on this line is simple. For instance:

• The number \(1\) is placed to the right of zero.
• A number like \(-2\) is found on the left side of zero.
• For decimal numbers like \(5.33\), it's positioned even further to the right of \(1\).

Understanding the concept of the number line is vital when learning about inequalities, as it graphically shows which number is greater or smaller by their positional order.

Decimal Conversion

Decimal conversion is the process of changing a fraction into a decimal form. This is especially useful when comparing numbers on the number line, as it allows for an easier and more intuitive understanding.
To change a fraction to a decimal, you perform division. For instance, the fraction \(\frac{16}{3}\) means you divide \(16\) by \(3\). This operation results in approximately \(5.33\), a value much simpler to locate and compare on a number line than its original fractional form.
Remember that some fractions result in terminating decimals, like \(\frac{1}{2} = 0.5\), while others lead to repeating decimals, as seen in \(\frac{16}{3} = 5.33\ldots\). Understanding these distinctions makes it easier to work with and visualize numbers when graphing or solving mathematical problems.