Problem 61 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ -10 \quad-100 $$

Step-by-Step Solution

Answer

-10 > -100.

Step 1: Identify the numbers

The numbers given in the problem are -10 and -100.

Step 2: Understand the comparison

We need to compare the two numbers, -10 and -100, to determine which is greater or if they are equal. Remember, on the number line, greater numbers are to the right and smaller numbers are to the left.

Step 3: Compare using the number line

On the number line, -10 is to the right of -100 because it is closer to zero, meaning -10 is greater than -100.

Step 4: Insert the correct symbol

Since -10 is greater than -100, we insert the 'greater than' symbol \(>\) to make the statement true.

Key Concepts

Number Line

When learning about integers, the number line is a wonderfully visual tool. It helps us to see how numbers are ordered. Imagine a straight, horizontal line with numbers placed at regular intervals. This is the number line.

• To the right: Numbers get larger as you move to the right. Positive numbers dominate this side.
• To the left: Numbers decrease in value. This side is where the negative numbers reside.

Think about the number line as a way to measure distance from zero. The further right you go, the larger the number's magnitude, and similarly, moving left reduces the size.
White space placement helps locate the numbers easier, and zero is always in the middle acting as a pivot for positive and negative numbers.

Integer Comparison

Now that we understand the number line, comparing integers becomes easier. Comparing simply means seeing which number is bigger, smaller, or if they are equal.

• Symbols such as < (less than), > (greater than), and = (equal to) are used to compare two numbers.

When we compare two numbers:

• Always remember that any number located to the right on the number line is larger than numbers to its left.
• If the numbers are directly on top of each other, they are equal.

Using the number line visually illustrates how -10 is greater than -100. Because -10 is nearer to zero, proving it's on the right side compared to -100.

Negative Numbers

Dealing with negative numbers can initially seem tricky. However, using the number line makes things much simpler. Negative numbers are just those that sit on the left of zero.

• With negative numbers, remember: the "larger" it is, the closer it will be to zero.
• For example, -5 is greater than -10 since -5 is closer to zero on the number line.
• Even though it feels opposite, because -100 is far from zero, it's actually smaller than numbers like -10 or -20.

Seeing these numbers in out everyday life, such as negative temperatures or elevations below sea level, helps relate to them in practical scenarios. They act just like positive numbers but find their place on the opposite side of zero.