Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$\frac{3}{4} \text { and }-\frac{5}{6}$$

Short answer

\(-\frac{5}{6}<\frac{3}{4}\) and \(\frac{3}{4}>-\frac{5}{6}\)

Step-by-step solution

Representation on the number line.

Since \(-\frac{5}{6}\) is a negative fraction, it is to the left of 0 on the number line; \(\frac{3}{4}\) is a positive fraction and it is to the right of 0. Therefore, first position -5/6 on the left side of 0 and then position 3/4 on the right side of 0.

Comparisons using inequalities

Now, we know from visual representation on the number line that \(-\frac{5}{6}\) is less than \(\frac{3}{4}\) because it is to the left of \(\frac{3}{4}\). So we can write two inequalities based on this: \(-\frac{5}{6}<\frac{3}{4}\) and \(\frac{3}{4}>-\frac{5}{6}\).

Final review of inequalities

Now, check the inequalities, they show that \(-\frac{5}{6}\) is less than \(\frac{3}{4}\) and \(\frac{3}{4}\) is greater than \(-\frac{5}{6}\), which matches our number line representation, hence our inequalities are correct.

Key concepts

Comparing Fractions

Understanding how to compare fractions is critical for accurately representing them on a number line and solving inequalities. When comparing fractions like \(\frac{3}{4}\) and \(\frac{5}{6}\), you should consider both their numerators and denominators. If the denominators are different, finding a common denominator can be helpful.

However, the task simplifies when comparing a positive and a negative fraction. A positive fraction is always greater than a negative fraction. We can say with certainty that \(\frac{3}{4}\) is greater than \(\frac{-5}{6}\) without calculating a common denominator. This is an essential rule when handling positive and negative numbers and impacts how we graph and interpret inequalities on a number line.

Inequalities

Inequalities express the relation between numbers, indicating which is larger or smaller. The inequality signs \( < \) and \( > \) are fundamental in mathematics for this purpose. For the exercise given, \(\frac{3}{4} > -\frac{5}{6}\) because \(\frac{3}{4}\) is on the right side of the number line when compared to \(\frac{-5}{6}\).

The sign \( > \) indicates that the first number is greater than the second, and the sign \( < \) indicates the opposite. It is crucial always to read inequalities from left to right—the way you would read a sentence—to understand the relationship between the numbers correctly.

Number Line Representation

The number line is a visual tool that helps us understand the order and spacing of numbers. The positions of \(\frac{3}{4}\) to the right of zero and \(\frac{-5}{6}\) to the left reflect their positive and negative values, respectively.

To properly place fractions on a number line, you identify benchmark numbers like 0, 1, or -1 and then determine where the fractions fall relative to these benchmarks. The fractions are graphed based on their value; for example, \(\frac{3}{4}\) is three-fourths of the way from 0 to 1. Proper placement on the number line allows for a straightforward comparison of their size and informs which inequality symbol to use.

Positive and Negative Fractions

Dealing with positive and negative fractions involves understanding that any positive fraction is to the right of zero on the number line, while negative fractions are to the left. This positioning is essential for quickly assessing which of two fractions is greater or lesser.

In the given exercise, \(\frac{3}{4}\) is a positive fraction and will be placed to the right side of 0, and \(\frac{-5}{6}\), being a negative fraction, will be placed to the left of 0. This distinction between positive and negative is critical for interpreting their values and is particularly useful when graphing numbers and evaluating inequalities, as well as understanding the overall concept of number positioning in mathematics.