Problem 23 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$-\frac{1}{2}\quad-\frac{3}{4}$$

Step-by-Step Solution

Answer

\(-\frac{1}{2} > -\frac{3}{4}\).

Step 1: Understand the Number Line

Visualize or draw a number line that includes negative fractions. Recall that numbers farther to the left on the number line are smaller than those to the right.

Step 2: Compare Magnitudes

Compare the absolute values of \(-\frac{1}{2}\) and \(-\frac{3}{4}\). The absolute value of \(-\frac{1}{2}\) is \(\frac{1}{2}\) and for \(-\frac{3}{4}\) it is \(\frac{3}{4}\). Since \(\frac{1}{2} < \frac{3}{4}\), \(-\frac{1}{2}\) is closer to zero than \(-\frac{3}{4}\).

Step 3: Select the Inequality Symbol

Because \(-\frac{1}{2}\) is closer to zero, it is greater than \(-\frac{3}{4}\). Thus, the correct symbol is \(>\).

Key Concepts

Negative Fractions

Negative fractions, like negative numbers, represent values less than zero. Think of them as the opposite of positive fractions. When dealing with negative fractions, it’s essential to remember the general rule: the more negative a number is, the smaller it is.

For example, let’s consider two negative fractions:

• \(-\frac{1}{2}\)
• \(-\frac{3}{4}\)

Which is larger? Since the fraction with a larger absolute value is further from zero and further left on a number line,

• \(-\frac{3}{4}\) has an absolute value of \(\frac{3}{4}\)
• \(-\frac{1}{2}\) has an absolute value of \(\frac{1}{2}\)

\(\frac{3}{4} > \frac{1}{2}\), therefore, \(-\frac{3}{4}\) is less than \(-\frac{1}{2}\).

To determine which negative fraction is larger, always look at which is closer to zero.

Number Line

The number line is a fundamental concept in understanding fractions, especially negative ones. It visually displays numbers in increasing order from left to right. When you plot values on a number line, negative fractions appear to the left of zero, and positive fractions appear to the right.

For instance,

• \(-\frac{1}{2}\), \(0\), and \(\frac{1}{2}\) are examples seen on a basic line.
• Fractions like \(-\frac{3}{4}\) would appear further left than \(-\frac{1}{2}\).

This alignment helps us easily identify which numbers are greater or smaller.When comparing two negative fractions using a number line:

• The fraction that appears further to the right is larger.
• Conversely, the one further left is smaller.

Thus, understanding placement on a number line is crucial for comparing negative fractions.

Absolute Value

The concept of absolute value refers to the distance a number is from zero on the number line, regardless of direction. It measures magnitude without regard to sign. For example, both \(-4\) and \(4\) have an absolute value of \(4\).

In the case of fractions:

• The absolute value of \(-\frac{1}{2}\) is \(\frac{1}{2}\).
• The absolute value of \(-\frac{3}{4}\) is \(\frac{3}{4}\).

Using absolute value allows us to compare the sizes of negative numbers and fractions. By ignoring the "negative" aspect, we focus on strength or size.In practical use:

• Compare absolute values to understand which negative fraction is "bigger" in terms of proximity to zero.
• A smaller absolute value indicates a number closer to zero.

This is why, when comparing \(-\frac{1}{2}\) and \(-\frac{3}{4}\), the absolute values are utilized to measure their sizes and placements on the number line effectively.