Problem 303 - Intermediate Algebra

Question

In the following exercises, locate the numbers on a number line. $$ \frac{3}{4},-\frac{3}{4}, 1 \frac{2}{3},-1 \frac{2}{3}, \frac{5}{2},-\frac{5}{2} $$

Step-by-Step Solution

Answer

Place \(\frac{3}{4},-\frac{3}{4}, 1 \frac{2}{3},-1 \frac{2}{3}, \frac{5}{2},-\frac{5}{2}\) at the respective positions on the number line.

Step 1: Identify the Numbers

List all the given numbers and identify their positions in terms of positive and negative values: Positive values: 1. \(\frac{3}{4}\) 2. 1 \(\frac{2}{3}\) 3. \(\frac{5}{2}\) Negative values: 1. -\(\frac{3}{4}\) 2. -1 \(\frac{2}{3}\) 3. -\(\frac{5}{2}\)

Step 2: Locate Positive Numbers on the Number Line

Start with the positive values: 1. \(\frac{3}{4}\) is between 0 and 1. Divide the segment between 0 and 1 into 4 equal parts and mark 3 parts. 2. 1 \(\frac{2}{3}\) is between 1 and 2. Divide the segment between 1 and 2 into 3 equal parts and mark 2 parts. 3. \(\frac{5}{2}\) is greater than 2. Convert to mixed number form: 2 \(\frac{1}{2}\). It is located halfway between 2 and 3.

Step 3: Locate Negative Numbers on the Number Line

Now, place the negative values: 1. -\(\frac{3}{4}\) is between 0 and -1. Divide the segment between 0 and -1 into 4 equal parts and mark 3 parts in the negative direction. 2. -1 \(\frac{2}{3}\) is between -1 and -2. Divide the segment between -1 and -2 into 3 equal parts and mark 2 parts in the negative direction. 3. -\(\frac{5}{2}\) is less than -2. Convert to mixed number form: -2 \(\frac{1}{2}\). It is located halfway between -2 and -3.

Step 4: Visualize on the Number Line

Draw a number line and place the points accordingly: -3, -2, -1, 0, 1, 2, 3Locate each of the identified points according to the previous steps.

Key Concepts

fractions

Fractions represent a part of a whole. They consist of two parts: the numerator (top number) and the denominator (bottom number).
The numerator tells you how many parts you have, while the denominator tells you into how many parts the whole is divided. For example, in \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator, meaning we have 3 out of 4 parts of a whole.
When locating fractions on a number line:

• First, identify the whole numbers between which the fraction lies.
• For \(\frac{3}{4}\), it lies between 0 and 1.
• Divide the segment between these two whole numbers into equal parts based on the denominator.
• Mark the fraction at the corresponding position.

This helps you visualize how fractions relate to whole numbers.

positive and negative numbers

Numbers can be either positive or negative, showing their direction relative to zero on a number line.
Positive numbers are greater than zero and are found to the right of zero. Negative numbers are less than zero and are found to the left of zero. For example, \(\frac{3}{4}\), 1 \(\frac{2}{3}\), and \(\frac{5}{2}\) are positive because they are greater than zero, while -\(\frac{3}{4}\), -1 \(\frac{2}{3}\), and -\(\frac{5}{2}\) are negative because they are less than zero.
Tips for working with positive and negative numbers:

• When dealing with negative fractions or mixed numbers, disregard the negative sign initially.
• Place the number on the positive side, then move it to the corresponding negative side.
• For example, locate \(\frac{3}{4}\) on the positive side, then plot -\(\frac{3}{4}\) directly opposite it on the negative side.

By following these steps, you can accurately place both positive and negative numbers on a number line.

mixed numbers

Mixed numbers combine whole numbers and fractions. They are written as a whole number followed by a fraction, like 1 \(\frac{2}{3}\).
To better understand mixed numbers:

• Identify the whole number part first, then focus on the fractional part.
• For example, in 1 \(\frac{2}{3}\): 1 is the whole number, and \(\frac{2}{3}\) is the fractional part.
• To locate mixed numbers on a number line, first find the whole number part, then place the fraction.

If needed, you can also convert mixed numbers to improper fractions to simplify plotting them on a number line. For instance, 1 \(\frac{2}{3}\) becomes \(\frac{5}{3}\), which helps to precisely divide the number line segment.
Mixed numbers offer a way to visualize quantities greater than one and help to easily place values on a number line. They blend the simplicity of whole numbers with the precision of fractions, making them essential for understanding number relationships.