Problem 22 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$5.7 \text { and }-4.2$$

Step-by-Step Solution

Answer

The two inequalities that compare 5.7 and -4.2 are \(5.7 > -4.2\) and \(-4.2 < 5.7\).

Step 1: Create the number line and place the numbers

Draw a number line and mark the positions of 5.7 and -4.2. It's essential to remember numbers to the right are greater than those to the left.

Step 2: Comparing the numbers

The number 5.7 is to the right of -4.2, which means 5.7 is greater than -4.2.

Step 3: Writing the inequalities

The two inequalities comparing the numbers are \(5.7 > -4.2\) and \(-4.2 < 5.7\). These mean the same thing but are expressed differently, the first one reads '5.7 is greater than -4.2', the second one reads '-4.2 is less than 5.7'.

Key Concepts

Number Line

A number line is a visual tool that helps us understand the position of numbers in relation to each other. Imagine it as a horizontal line with evenly spaced markings, each representing a number. This line extends infinitely in both directions: to the right for positive numbers and to the left for negative numbers.
When using a number line, we place each number on a specific point to represent its value. For instance, in our exercise, we can place 5.7 on the right side of zero, and -4.2 on the left side. This helps us quickly identify the relative sizes of the numbers.

• Numbers to the right are greater than those to the left.
• Zero is a helpful reference point; positive numbers are to its right, negatives to its left.

Having this clear visual can make tasks like comparing numbers and understanding inequalities much more straightforward.

Comparing Numbers

Comparing numbers involves deciding which of the two numbers is larger or smaller. It's a handy concept in mathematics, helping us to organize numbers and solve problems systematically.
On the number line, the number that appears further to the right is always greater. Conversely, the one further to the left is smaller. In the example of 5.7 and -4.2, 5.7 is located to the right, showing it is the larger number. On the flip side, -4.2 being to the left indicates it's the smaller number.

• Use the position on the number line to easily compare numbers.
• Right is greater, left is lesser.

Comparing numbers also helps us in writing inequalities, which express these relationships in mathematical terms.

Inequality Notation

Inequality notation is how we write about relationships between numbers, indicating whether one is greater than or less than another. This notation uses symbols to make clear the size relationship between two numbers. There are two primary symbols:

• "\(>\)" which means "greater than"
• "\(<\)" which means "less than"

For our example, we express the larger number 5.7's relationship with -4.2 using these symbols. We say "5.7 is greater than -4.2," noted as \(5.7 > -4.2\). Conversely, "-4.2 is less than 5.7," noted as \(-4.2 < 5.7\).
Inequality notation is a concise and informative way to describe the positions of numbers on the number line relative to each other. Understanding and writing inequalities is a key mathematical skill that clearly communicates how numbers compare.