Problem 23 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-2.8 \text { and } 3.7$$

Step-by-Step Solution

Answer

The two inequalities comparing -2.8 and 3.7 are \( -2.8 < 3.7 \) and \( 3.7 > -2.8 \).

Step 1: Identify the Numbers

The numbers to be plotted on the number line are -2.8 and 3.7.

Step 2: Plot the Numbers on the Number Line

Draw a number line. On this line, -2.8 is placed to the left because it is smaller, and 3.7 is placed to the right because it is larger.

Step 3: Write Inequalities

Notice that -2.8 is less than 3.7 and 3.7 is greater than -2.8. So the two inequalities are \( -2.8 < 3.7 \) and \( 3.7 > -2.8 \).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that compare two numbers or expressions. They show the relationship between values using symbols like < (less than) or > (greater than). When dealing with numbers on a number line, inequalities help us see which number is smaller or larger. For example, if you have two numbers, such as -2.8 and 3.7, you can use inequalities to compare them:

• If one number is less than the other, you use the "<" symbol. In this case, -2.8 is less than 3.7, so it is written as \(-2.8 < 3.7\).
• Conversely, if one number is greater, you use the ">" symbol. Here, 3.7 is greater than -2.8, so it becomes \(3.7 > -2.8\).

Understanding and using inequalities help us compare numbers quickly, showing us which is larger or smaller without needing to measure them physically on a number line. These symbols provide a handy shortcut in expressing number relationships.

Graphing Numbers on a Number Line

A number line is a straight line with numbers placed at equal intervals along its length. It's an excellent tool for graphing numbers and visually comparing their sizes. To graph a number on this line, you simply mark its position relative to zero and other numbers. To graph -2.8 and 3.7:

• First, draw a horizontal line and label points evenly spanning from at least the smallest to the largest numbers you are graphing. For our example, the line should extend from below -2.8 to above 3.7.
• Next, locate -2.8, remembering it will be to the left of zero since it's a negative number.
• Then, find 3.7, which is to the right of zero given it's positive.

Number lines offer a simple but powerful way to see how numbers relate to one another. They clarify the concept of distance and sequence by showing exactly where numbers sit in relation to each other and zero.

Comparing Numbers

Comparing numbers is all about determining their size in relation to one another. It involves deciding which number is greater, lesser, or if the two numbers are equal. This comparison is crucial in many mathematical operations and can simplify solving real-world problems. When comparing -2.8 and 3.7:

• Notice that negative numbers are always smaller than positive numbers. Therefore, when comparing a negative and a positive number, like -2.8 and 3.7, the negative number will always be less.
• To compare two positive numbers, or two negative numbers, you need to see which one lies further to the right on a number line. The further right, the larger the number.

Understanding how to compare numbers using a number line and inequalities aids in solving math problems efficiently. It is an essential skill that can be used in various areas, from basic arithmetic to advanced algebra.