Problem 60 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -11 \quad-12 $$

Step-by-Step Solution

Answer

-11 > -12

Step 1: Understand the Symbols

In this exercise, the symbols \(<\) and \(>\) are used to compare two numbers. \(a < b\) means that \(a\) is less than \(b\), while \(a > b\) means that \(a\) is greater than \(b\).

Step 2: Compare the Numbers

The numbers given are \(-11\) and \(-12\). To determine which symbol to use, we need to evaluate how these numbers compare on a number line. Remember, on a number line, a number is greater if it is to the right of another number.

Step 3: Place -11 and -12 on the Number Line

Both \(-11\) and \(-12\) are negative numbers. Since -11 is to the right of -12 on the number line, -11 is actually greater than -12.

Step 4: Insert the Correct Symbol

Based on Step 3, \(-11\) is greater than \(-12\). So, we insert the \(>\) symbol between the numbers to make the statement \(-11 > -12\) true.

Key Concepts

Comparing Negative Numbers

When it comes to comparing negative numbers, the process can seem a bit counterintuitive at first. With positive numbers, larger numbers have greater values, but the opposite is true for negative numbers. Here’s how it works:

• Negative numbers are those less than zero, represented with a minus sign, such as -1, -2, -10, etc.
• A negative number is smaller if its value is further from zero. For example, -12 is less than -5 because it is further left on the number line.

This means that -12 is smaller than -11, even though 12 is larger than 11 when we talk about positive numbers. In the real world, think of negative numbers in terms of debt: owing 12 dollars is less favorable than owing 11, hence -12 is less than -11. Therefore, remember that in negative numbers, the smaller the absolute value, the larger the number.

Number Line Concepts

The number line is a straight line where numbers are placed in their correct position relative to each other. Understanding number line concepts can greatly aid in visualizing and comparing numbers, including negative ones. Here’s what to keep in mind:

• The number line is usually drawn horizontally with numbers placed in increasing value from left to right.
• Zero is the central dividing point between positive and negative numbers.
• Negative numbers are to the left of zero, and as you move further left, the numbers decrease in value.

By visualizing numbers on a line, you can easily determine which number is greater or lesser. For example, -11 is to the right of -12, which makes it greater. This spatial awareness helps clarify comparisons, particularly with negative numbers. You can grasp that numbers further to the right are always greater than those to the left.

Ordering Real Numbers

Ordering real numbers involves arranging them from smallest to largest or vice versa. This applies to both positive and negative numbers, as well as decimals and fractions. Here’s a guide to ordering real numbers effectively:

• First, separate your positive and negative numbers because they fall into entirely different realms: positives are always greater than any negative.
• Within the negative group, remember that numbers closer to zero are larger.
• Incorporate any decimal or fractional values by converting them to similar forms for easy comparison.

In practice, consider an example listing: -12, -11, 0, 3.5, 9. Begin with the negative numbers. Between -11 and -12, the closer to zero (-11) is greater. Following zero, continue with positive numbers which naturally follow in ascending order. Ordering real numbers is essentially stacking them ricochet style across the number spectrum like beads on a string, taking careful note of their values in relation to zero and other numbers.