Problem 22 - A Graphical Approach to College Algebra

Question

Graph each set of numbers on a number line. $$\left\\{-0.6, \frac{9}{8}, 2.5, \frac{13}{4}\right\\}$$

Step-by-Step Solution

Answer

Convert numbers to decimal form and plot them on a number line from \(-1\) to \(4\).

Step 1: Convert To Decimal Form

It's helpful to express all numbers in decimal form for easy comparison and plotting on the number line. The given numbers are \(-0.6\), \(\frac{9}{8}\), \(2.5\), and \(\frac{13}{4}\). Convert the fractions to decimals: \(\frac{9}{8} = 1.125\) and \(\frac{13}{4} = 3.25\). Now, the decimal equivalents are \(-0.6\), \(1.125\), \(2.5\), and \(3.25\).

Step 2: Determine Range of the Number Line

Identify the largest and smallest numbers to establish the range of our number line. The smallest value is \(-0.6\), and the largest is \(3.25\). Ensure that the number line extends at least from \(-1\) to \(4\) to comfortably include all the given numbers.

Step 3: Mark Key Points on Number Line

Draw a horizontal line. Mark integer values on the line for easy plotting: \(-1, 0, 1, 2, 3,\) and \(4\). These points help in accurately placing non-integers.

Step 4: Plot Each Decimal on the Number Line

Locate each decimal on the number line:- \(-0.6\) is slightly before \(0\).- \(1.125\) is slightly past \(1\) but before \(1.5\).- \(2.5\) is exactly halfway between \(2\) and \(3\).- \(3.25\) is quarter-way past \(3\) but before \(3.5\). Place points on the line representing these positions.

Key Concepts

Number Line

A number line is a visual representation of numbers positioned on a straight line. It helps us understand the relative distance between different numbers.
Number lines are typically horizontal, with numbers placed at regular intervals. These intervals can represent whole numbers or fractions. A number line begins with a zero point at its center, with negative numbers to the left and positive numbers to the right.

• This visual tool helps in comparing the magnitude of numbers.
• It is a useful method for visualizing operations like addition, subtraction, and locating decimal points or fractions.

When graphing decimal numbers and fractions, it creates a clear visual comparison of their sizes and positions.

Decimal Conversion

Decimal conversion is the process of changing a number from another form (like a fraction) into a decimal form. Decimal form is a way of expressing numbers using base 10, where each digit represents a power of 10.
This form is useful for easily comparing and calculating with non-integer numbers. For example, converting numbers into decimals simplifies locating them on a number line or performing arithmetic operations.

• To convert a fraction into a decimal, divide the numerator by the denominator. For instance, \(\frac{9}{8} = 1.125\) is obtained by dividing 9 by 8.
• Decimals provide a straightforward method to express quantities that are not whole numbers.

Thus, decimal conversion plays a crucial role in simplifying numerical tasks.

Fraction to Decimal Conversion

Converting fractions to decimals is an essential skill in mathematics, allowing easier computation and comparison of values. The process of fraction to decimal conversion involves dividing the numerator (top number) by the denominator (bottom number) using long division.
For example, to convert \(\frac{9}{8}\) into a decimal:

• Divide 9 by 8, which results in 1.125, meaning \(\frac{9}{8} = 1.125\).
• This conversion helps in visualizing the number on the number line as it can be more intuitive for both positive and negative fractions.

Understanding this process is fundamental when working with different types of numbers, particularly in activities like plotting points.

Plotting Points

Plotting points on a number line involves placing dots on the precise location where a number exists relative to others.
These points are determined based on the decimal form of each number. Accurate plotting requires an understanding of the number line's scale.

• Identify each number's position based on its decimal value. For example, \(-0.6\) is located to the left of zero.
• Mark clear and distanced points like an interval to distinguish between the values visually.

Such an approach facilitates clear visualization of number sizes, aiding in tasks like graphing sets of numbers or analyzing differences.