Problem 45 - Elementary Algebra

Question

Fill in the blank with \(<,=,\) or \(\geqslant\). -0.5 _____ -1.5

Step-by-Step Solution

Answer

-0.5 > -1.5

Step 1: Understand the Numbers

First, recognize the two numbers involved: -0.5 and -1.5. These are both negative numbers.

Step 2: Recall the Number Line

Think about a number line, where numbers grow smaller as you go to the left and larger as you go to the right. In negative numbers, a smaller negative number closer to zero is actually a larger value.

Step 3: Compare the Numbers

Since -0.5 is closer to zero than -1.5 on the number line, -0.5 is actually greater than -1.5.

Step 4: Choose the Correct Symbol

We need to choose the symbol that indicates -0.5 is greater than -1.5. In this case, the symbol is > (greater than).

Key Concepts

Absolute Value

When discussing numbers, especially negative ones, it becomes essential to understand the concept of absolute value. Absolute value is simply the distance a number is from zero on the number line, regardless of direction.
For instance, the absolute value of both -5 and 5 is the same, which is 5, because each is five units away from zero. This helps because it allows you to see numbers stripped of their signs and focus on their magnitude.
To denote the absolute value of a number, we use vertical bars like this:

• Absolute value of -0.5 is \(|-0.5| = 0.5\).
• Absolute value of -1.5 is \(|-1.5| = 1.5\).

With absolute values, you can compare the sizes of numbers without worrying about negative or positive signs. Remember, absolute values are always non-negative.

Number Line

A number line is a straightforward, visual way to understand numbers and their relationships.
You imagine it as a horizontal line where each point corresponds to a number. Zero is typically in the middle, with positive numbers extending to the right and negative numbers to the left.
When comparing numbers, especially negatives, envisioning or drawing a number line can be extremely helpful:

• A number closer to the right is larger and has a greater value than a number further to the left.
• When considering negative numbers, such as -0.5 and -1.5, -0.5 is to the right of -1.5, making -0.5 larger.

Understanding the number line is beneficial for visualizing inequalities and comparisons effectively.

Inequality Symbols

Inequality symbols are essential in mathematics as they allow us to express the relationship between two values:

• "\(<\)" means less than: Used when the number on the left is smaller than theone on the right.
• "\(>\)" signifies greater than: Used when the number on the left is larger than theone on the right.
• "\(=\)" indicates that both numbers are equal.
• "\(\leqslant\)" and "\(\geqslant\)" mean less than or equal to, and greater than or equal to, respectively.

For the specific example of comparing -0.5 and -1.5, we conclude that -0.5 is greater than -1.5. Thus, we use the greater than symbol:
-0.5 \(>\) -1.5.
Understanding these symbols makes it clearer to express and solve mathematical relationships effectively.