Problem 44 - Algebra A Combined Function

Question

Graph each set of numbers on the number line. $$ 4.5,-\frac{9}{4}, 1.75,-\frac{7}{2} $$

Step-by-Step Solution

Answer

Convert to decimals, order them, and plot on the number line: -3.5, -2.25, 1.75, 4.5.

Step 1: Convert All Numbers to Decimals

First, we need to convert all fractions into decimal form for easier plotting. The number \(-\frac{9}{4}\) can be converted to decimal by dividing 9 by 4, which gives \(-2.25\). Similarly, \(-\frac{7}{2}\) is \(-3.5\). So, now we have the decimals: 4.5, -2.25, 1.75, and -3.5.

Step 2: Determine the Order on the Number Line

Arrange the numbers in order from least to greatest so we can correctly position them on the number line. The sorted order is: \(-3.5, -2.25, 1.75, 4.5\).

Step 3: Sketch the Number Line

Draw a horizontal line to represent the number line. Mark evenly spaced intervals and label them with integers that cover the range of the numbers: -4 to 5, for instance, since our numbers range from \(-3.5\) to 4.5.

Step 4: Plot Each Number on the Number Line

Plot each number on the number line by marking a point along the line. Place \(-3.5\) to the left of \(-3\), \(-2.25\) slightly before \(-2\), 1.75 between 1 and 2, and 4.5 between 4 and 5.

Key Concepts

Decimal Conversion

When dealing with numbers, especially on a number line, converting fractions to decimals can make the process easier. This conversion helps in visual comparisons and simplifies plotting. To convert a fraction like \(-\frac{9}{4}\) into a decimal, divide the numerator (9) by the denominator (4). In this case, it results in \(-2.25\). Similarly, we convert \(-\frac{7}{2}\) by dividing 7 by 2, which gives us \(-3.5\). By having all numbers in decimal form, it becomes much simpler to compare and plot them accurately on the number line.

Ordering Numbers

Once you have your decimals, ordering these numbers from smallest to largest is crucial for plotting. This lets you arrange numbers correctly on the number line. For example, with the decimals \(-3.5, -2.25, 1.75,\) and \(4.5\), the logical sequence from least to greatest is: \(-3.5, -2.25, 1.75,\) and \(4.5\). This step ensures clarity and precision, dictating where each point should be plotted along the line. Remember that negative numbers are smaller than positive numbers, and larger negative numbers are placed further left on the number line.

Plotting Points

With your ordered list, you can now focus on plotting these points on the number line. Draw a horizontal line, ensuring it has the necessary range, here from at least \(-4\) to \(5\). Each number is plotted at a specific position:

• \(-3.5\) slightly left of \(-3\).
• \(-2.25\) before \(-2\).
• \(1.75\) between \(1\) and \(2\).
• \(4.5\) between \(4\) and \(5\).

By spacing these intervals evenly, you ensure that your line accurately represents numerical relationships and distances.

Fractions to Decimals

To successfully navigate number lines, converting fractions to decimals is often a first step. It simplifies visualization and comparison. Consider fractions like \(-\frac{9}{4}\) and \(-\frac{7}{2}\). Converting \(-\frac{9}{4}\) to decimal form yields \(-2.25\), and \(-\frac{7}{2}\) converts to \(-3.5\). Use division for this process: divide the top number by the bottom. Decimal conversions make aligning these numbers on a number line much more intuitive. Remember, a decimal provides a straightforward way to express exactly where to place a fraction numerically, especially in exercises involving multiple number positions.