To determine whether [tex]\(-\frac{5}{6}\)[/tex] is less than or greater than [tex]\(-\frac{8}{9}\)[/tex], consider the properties of negative fractions and how they compare to each other on the number line.
### Step-by-Step Solution:
1. Understand the Position on the Number Line:
When comparing two negative fractions, the fraction with the smaller absolute value (ignoring the negative sign) represents a number closer to zero and is therefore the greater of the two numbers. Conversely, the fraction with the larger absolute value represents a number farther from zero and thus is smaller when considering negative values.
2. Compare Absolute Values:
Let's compare the absolute values of the given fractions:
- The absolute value of [tex]\(-\frac{5}{6}\)[/tex] is [tex]\(\frac{5}{6}\)[/tex].
- The absolute value of [tex]\(-\frac{8}{9}\)[/tex] is [tex]\(\frac{8}{9}\)[/tex].
3. Find a Common Denominator:
To compare [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{8}{9}\)[/tex] directly, we can convert them to have a common denominator. The least common multiple (LCM) of 6 and 9 is 18.
- Converting [tex]\(\frac{5}{6}\)[/tex] to a denominator of 18:
[tex]\[
\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}
\][/tex]
- Converting [tex]\(\frac{8}{9}\)[/tex] to a denominator of 18:
[tex]\[
\frac{8}{9} = \frac{8 \times 2}{9 \times 2} = \frac{16}{18}
\][/tex]
4. Compare Converted Fractions:
Now that both fractions have the same denominator, we can compare the numerators:
[tex]\[
\frac{15}{18} \quad \text{and} \quad \frac{16}{18}
\][/tex]
Since [tex]\(15 < 16\)[/tex], it implies:
[tex]\[
\frac{15}{18} < \frac{16}{18}
\][/tex]
5. Interpret the Comparison for Negative Values:
Since [tex]\(\frac{5}{6} < \frac{8}{9}\)[/tex], the negative versions of these fractions will have the reverse inequality:
[tex]\[
-\frac{5}{6} > -\frac{8}{9}
\][/tex]
Therefore, the correct symbol to insert in the statement [tex]\(-\frac{5}{6} \quad -\frac{8}{9}\)[/tex] is [tex]\(>\)[/tex]:
[tex]\[
-\frac{5}{6} > -\frac{8}{9}
\][/tex]
