Problem 22 - A Graphical Approach to Precalculus with Limits

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 fractions to decimals, then plot all values on a number line from -1 to 4.

Step 1: Convert Fractions to Decimals

To make it easier to plot the numbers on a number line, first convert all fractions to decimal form. Let's start with \(\frac{9}{8}\):\[\frac{9}{8} = 1.125\]Now, convert \(\frac{13}{4}\):\[\frac{13}{4} = 3.25\]

Step 2: List All Numbers in Decimal Form

List all the numbers in decimal form, including the ones already given in decimal:- \(-0.6\)- \(1.125\) (from \(\frac{9}{8}\))- \(2.5\)- \(3.25\) (from \(\frac{13}{4}\))

Step 3: Determine Range for Number Line

Determine the range of the number line by looking at the smallest and largest values in your list of decimals. The smallest number is \(-0.6\) and the largest is \(3.25\), so our number line should range from at least \(-1\) to \(4\) to comfortably include all values.

Step 4: Draw Number Line

Draw a horizontal line. Mark key values along the line: evenly space integers like \(-1, 0, 1, 2, 3,\) and \(4\), considering the decimals are in between these points.

Step 5: Plot Each Decimal on the Number Line

Plot each of the decimal numbers on your number line:- \(-0.6\) is just over halfway between \(-1\) and \(0\).- \(1.125\) is slightly past \(1\).- \(2.5\) is exactly halfway between \(2\) and \(3\).- \(3.25\) is a quarter of the way past \(3\).

Key Concepts

Converting Fractions to Decimals

Understanding how to convert fractions into decimals is crucial for many mathematics problems, such as graphing numbers on a number line. A fraction represents a division problem, where the top number (numerator) is divided by the bottom number (denominator). For example, if you have the fraction \(\frac{9}{8}\), you can convert it by dividing 9 by 8. Using division, \(9 \div 8 = 1.125\). Similarly, for \(\frac{13}{4}\), divide 13 by 4 to get 3.25.
For better results with manual calculations, you might find it helpful to use long division:

• Place the numerator (the number on top) inside the division bracket.
• Place the denominator (the number on bottom) outside the bracket.
• Perform the division to obtain the decimal.

This method will help you convert any fraction to a decimal correctly, making it easy to work with numbers in different formats.

Decimal Representation

Once you have converted fractions into decimals, understanding their representation is next. Decimals are another form of representing fractions and are found to the right of a decimal point. In our example, \(-0.6\), \(1.125\), \(2.5\), and \(3.25\) are all numbers in decimal form. Notice how each number carries a specific place value:

• The first digit to the right of the decimal point is the tenths place.
• The second digit is the hundredths place.
• The third is the thousandths place, and so on.

This place value system helps you determine the exact position of each number on a number line. For example, \(2.5\) is halfway between 2 and 3 because the 5 is in the tenths place. Understanding decimal representation allows you to interpret the size and position of numbers more accurately in any context.

Plotting Numbers

Plotting numbers on a number line is a visual way to understand how different values relate to each other. After converting and listing all numbers as decimals, the next step is to draw a number line. This line should cover the range of numbers you need, from the smallest to the largest. Here, our numbers range from \(-0.6\) to \(3.25\), so as suggested, a number line from -1 to 4 is ideal.

Once the line is drawn, mark easy-to-see reference points, like whole numbers. Then you position each decimal in relation to these reference points:

• For \(-0.6\), place it a little past halfway between -1 and 0.
• For \(1.125\), position it slightly past 1.
• \(2.5\) fits right between 2 and 3.
• \(3.25\) should be placed a quarter past 3.

By plotting these points, you can visually grasp how close or far apart numbers are relative to one another, enhancing your understanding.