Problem 116 - Intermediate Algebra

Question

Place the proper symbol ( \(>\) or \(<\) ) in the blank: $$ -4.6 \quad \square\quad-4.5 $$

Step-by-Step Solution

Answer

-4.6 < -4.5

Step 1: Understand the Number Line

Both numbers, -4.6 and -4.5, are negative. On a number line, numbers increase in value as they move to the right and decrease as they move to the left.

Step 2: Comparing the Numbers

Since both numbers are negative, the number with the smaller absolute value is actually greater. The absolute value of -4.6 is 4.6, while the absolute value of -4.5 is 4.5. Since 4.5 is less than 4.6, -4.5 is greater than -4.6.

Step 3: Place the Correct Symbol

Because -4.5 is greater than -4.6, we use the less than symbol (`<`) to show this relationship. The expression becomes: \[-4.6 < -4.5\]

Key Concepts

Absolute Value

Absolute value is a mathematical concept that refers to the distance of a number from zero on the number line, without regard to its direction. This means that both negative and positive numbers have the same absolute value if they are the same distance from zero.
For instance:

• The absolute value of -3 is 3.
• The absolute value of 3 is also 3.

The absolute value is always a non-negative number. For negative numbers, it reflects the magnitude without the negative sign. In problems involving inequality or locating positions on the number line, understanding absolute values helps determine which numbers are closer or further from zero. This can be particularly useful when comparing numbers, especially negative ones.

Negative Numbers

Negative numbers are numbers less than zero, and they are represented with a minus sign (-). They are the mirror counterpart of positive numbers on the number line.
Here are some key points about negative numbers:

• They are used to represent values less than zero, like debts or temperatures below freezing.
• On the number line, they appear to the left of zero.
• The more negative a number is, the smaller its value. For example, -5 is less than -2.

When dealing with negative numbers, it's helpful to visualize them on the number line, where they give a clearer sense of order and size compared to one another. Remember, when comparing negative numbers, the number with the smaller absolute value is larger.

Comparing Numbers

Comparing numbers, especially negatives, involves understanding their position and magnitude on the number line. When we compare two numbers, we determine which is larger or smaller.
For comparing negative numbers:

• The number closer to zero is larger. For example, -4.5 is greater than -4.6 because it is closer to zero.
• Use the concept of absolute value to help decide which is larger. The number with the smaller absolute value is actually larger when both numbers are negative.
• Symbols such as `>` and `<` show relationships: - `>` means greater than. - `<` means less than.

In our original exercise, although -4.6 seems smaller when looking at its digits, it is actually larger in magnitude (less negative) than a number like -4.5. This is why in the comparison of \(-4.6 \quad < \quad-4.5\). Understanding these comparisons accurately reveals the true relationship between numbers on a number line.