Problem 76 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ |-12| \quad \frac{24}{2} $$

Step-by-Step Solution

Answer

|-12| = \frac{24}{2}

Step 1: Evaluate the absolute value

Calculate the absolute value of \(-12\). The absolute value of a number is its distance from zero on the number line without considering the direction. Thus, \(|-12| = 12\).

Step 2: Calculate the fraction

Simplify the fraction \(\frac{24}{2}\) by dividing 24 by 2. This simplifies to 12. Therefore, \(\frac{24}{2} = 12\).

Step 3: Compare the two values

Now that we have \(|-12| = 12\) and \(\frac{24}{2} = 12\), compare these two numbers. Since both numbers are equal, we insert \(=\) between them.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator. The numerator is the top number which represents how many parts we have. The denominator is the bottom number which tells us how many equal parts the whole is divided into. For example, in the fraction \( \frac{24}{2} \), 24 is the numerator and 2 is the denominator.
When working with fractions, it's often useful to simplify them, which means writing them in the smallest possible equivalent form. Simplifying makes fractions easier to understand and work with.

• In our exercise, we simplified \( \frac{24}{2} \) by dividing both the numerator and the denominator by 2, resulting in 12.

Simplifying fractions is essential when making comparisons because it allows us to easily see relationships between different numbers.

Number Line

A number line is a simple way to visually represent numbers in order. It shows numbers as points on a line, which helps us easily see the size and distance between them.
The absolute value is closely related to number lines because it measures a number's distance from zero, regardless of direction. By finding the absolute value of a number, we can understand how far it is from zero on the number line.

• For instance, the absolute value of \(-12\) is 12, since it is 12 units away from zero on the number line.

A number line can help visualize fractions and other numbers. By placing them on the line, comparisons become more straightforward as we see which ones are larger or smaller by their positions.

Comparisons

Comparisons allow us to determine how two numbers relate to each other, which is particularly useful in mathematical operations and problem-solving. We use symbols like \(<\), \(>\), and \(=\) to express comparisons, indicating whether a number is less than, greater than, or equal to another.
In the provided exercise, we evaluated the absolute value of \(-12\), which is 12, and simplified the fraction \( \frac{24}{2} \), also 12.

• Since both results are the same, we use the \(=\) symbol.

Making effective comparisons relies on simplifying numbers wherever possible. Using absolute values and reducing fractions help make these comparisons clearer and more straightforward.
Understanding how to compare numbers properly is a fundamental skill, simplifying more complex mathematical operations and applications.