Problem 40 - Beginning and Intermediate Algebra

Question

Graph each number on a number line. \(5.25,4 \frac{5}{9},-2 \frac{1}{3}, 0,-3 \frac{2}{5}\)

Step-by-Step Solution

Answer

Plot the numbers at 5.25, 4.56, -2.33, 0, and -3.4 on the number line.

Step 1: Identify the Numbers

List the numbers to be graphed: \(5.25, 4 \frac{5}{9}, -2 \frac{1}{3}, 0, -3 \frac{2}{5}\).

Step 2: Convert Fractions to Decimals

Convert the mixed numbers into decimal form: \(5.25\) remains as it is. Next, \(4 \frac{5}{9} = 4 + \frac{5}{9} = 4 + 0.56 \approx 4.56\). Then, \(-2 \frac{1}{3} = -2 - \frac{1}{3} = -2 - 0.33 \approx -2.33\). Similarly, \(-3 \frac{2}{5} = -3 - \frac{2}{5} = -3 - 0.4 = -3.4\).

Step 3: Draw the Number Line

Draw a horizontal line and mark equal intervals on it. Label these intervals with appropriate values.

Step 4: Plot the Points

Locate each converted decimal number on the number line:- Place \(5.25\) slightly to the right of \(5\).- Place \(4.56\) slightly past the halfway point between \(4.5\) and \(5\).- Place \(-2.33\) slightly before the third division between \(-2\) and \(-3\).- Place \(0\) at the origin.- Place \(-3.4\) slightly past the midpoint between \(-3\) and \(-4\).

Key Concepts

Converting Fractions to Decimals

When working with fractions and mixed numbers, you often need to convert them into decimals to make graphing on a number line simpler.
To convert a mixed number to a decimal:

• Separate the whole number part from the fractional part.
• Convert the fractional part to a decimal.
• Add the decimal value to the whole number.

For example, in the fraction 4 5/9, you first take the whole number 4, and then you convert the fraction 5/9 into a decimal. To do this, you divide 5 by 9: 5 ÷ 9 ≈ 0.56. Therefore, 4 5/9 becomes 4 + 0.56 = 4.56. For negative mixed numbers like -2 1/3, the process is similar: -2 is the whole number, and 1/3 converted to decimal is approximately 0.33. So, -2 1/3 becomes -2 - 0.33 ≈ -2.33. Remember, keeping the signs (positive or negative) right is crucial for accurate plotting.

Plotting Points

Plotting numbers on a number line involves several steps to ensure clarity

• Draw the number line. A horizontal line with equal intervals marked to represent different values.
• Determine the scale. Choose equal spaces between each number, considering the numbers you need to plot to fit them comfortably on the line.

For instance, when plotting 5.25, find the 5 mark on the number line and move a quarter step to the right.
Similarly, for 4.56, locate 4.5, move slightly more than halfway between 4.5 and 5. With negative numbers like -2.33 and -3.4, locate the negative positions -2 and -3, then move left (because they are negative) by 0.33 and 0.4, respectively. Lastly, plot 0 at the origin. Always ensure the points are plotted with accuracy, reflecting their exact decimal positions.

Number Line

A number line is a fundamental concept in mathematics, representing numbers graphically.
It's a straight line where each point corresponds to a number. Here’s how to draw a number line and plot points:

• Draw a straight horizontal line.
• Mark equal intervals for each unit.
• Label the intervals according to the numbers you aim to represent.

For this exercise, you may start from -4 to 6 to accommodate all numbers in the list: 5.25, 4.56, -2.33, 0, -3.4.

Interval points don’t have to be only units; sometimes, you mark them as fractions or decimals for precision (e.g., marking 0.25 between 0 and 1). Accurate labeling helps in correctly plotting and interpreting the points on the number line.