Problem 40 - Prealgebra and Introductory Algebra

Question

Write the given numbers in order from smallest to largest. $$-6,2,-8,-7$$

Step-by-Step Solution

Answer

The given numbers in order from smallest to largest are \(-8, -7, -6, 2\).

Step 1: Identify the Numbers

First, Identify the numbers given in the problem: \(-6, 2, -8, -7\).

Step 2: Compare the Numbers

Compare the given numbers, keeping in mind that higher absolute values of negative numbers mean those numbers are smaller. From the numbers provided, \(-8\) has the highest absolute value among the negatives, so it is the smallest number. After \(-8\), the next smallest number would be \(-7\) and then \(-6\).

Step 3: Order the Numbers

Arrange the numbers in ascending order based on the prior findings. This gives us \(-8, -7, -6\) and finally \(2\) which is the largest number of the four presented.

Key Concepts

Negative Numbers

Negative numbers are values that are less than zero. They are typically used to represent a decrease or opposite quantity, such as a temperature below freezing or a financial loss. For example, if you have -3 dollars, it means you owe 3 dollars, or more simply, your balance is 3 dollars less than zero.

To visualize negative numbers, think of a number line: zero is at the center, positive numbers are to the right, and negative numbers are to the left. The further left you go, the smaller the value of the number. Therefore, -8 is less than -6 because -8 is further left on the number line than -6.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is always a non-negative value. Essentially, the absolute value of a number is the same number without its sign.

For instance, both -5 and 5 have an absolute value of 5, because both are five units away from zero. It’s important to remember that absolute value affects how we compare numbers.

• Even though -8 looks larger than -7 when considering their whole numbers, its absolute value is actually smaller.
• In practical problems, negative numbers with larger absolute values are smaller because they are farther from zero.

Comparing Integers

Comparing integers involves determining which of two or more numbers is greater or smaller. This process is crucial for ordering numbers correctly.

When you compare integers:

• Positive numbers are always greater than negative numbers.
• A larger number with a positive sign is greater than a smaller number with a positive sign.
• For negative numbers, the value with the smallest absolute value is actually the greater number because it is closer to zero.

This means that when you list integers from smallest to largest, you start with the most negative (the furthest left on the number line) and end with the most positive (the furthest right).

In the given problem, knowing these rules helps us confidently say that the correct order from smallest to largest is: -8, -7, -6, and finally 2.