Comparing Rational Numbers: -9/10 vs -3/10

The problem asks us to compare two rational numbers, and , and choose the correct sign to make the statement true. We are given three options: greater than ( ), less than ( ), and equal to ( ).

Step 1: Understand the numbers.
We are comparing two negative fractions. Both fractions have the same denominator, which is 10. This simplifies the comparison.

Step 2: Compare the numerators.
When comparing fractions with the same denominator, we compare their numerators. The numerators are -9 and -3.

Step 3: Determine the relationship between the numerators.
We need to determine whether -9 is greater than, less than, or equal to -3. On a number line, numbers to the right are greater, and numbers to the left are smaller. Since -3 is to the right of -9 on the number line, -3 is greater than -9. Therefore, .

Step 4: Apply the comparison to the fractions.
Since the denominators are positive and the same, the inequality of the numerators will be the same as the inequality of the fractions.
So, if , then .

Step 5: Choose the correct sign.
The statement means that is less than . The sign that represents “less than” is .

Step 6: Verify with intuition.
Imagine a number line. Zero is in the middle. Negative numbers are to the left of zero. The further a negative number is from zero, the smaller it is.
is units away from zero in the negative direction.
is units away from zero in the negative direction.
Since , is further from zero than in the negative direction. Therefore, is a smaller number than .

The final answer is .