Compare: `(12)/((-17))and(-6)/(7)`

Answer

To compare the rational numbers \( \frac{12}{-17} \) and \( \frac{-6}{7} \), we can follow these steps:

### Step 1: Identify the Rational Numbers
The given rational numbers are:
1. \( \frac{12}{-17} \)
2. \( \frac{-6}{7} \)

### Step 2: Rewrite the Rational Numbers
We can rewrite \( \frac{12}{-17} \) as \( -\frac{12}{17} \) to make it clearer that it is a negative number.

So, we have:
1. \( -\frac{12}{17} \)
2. \( \frac{-6}{7} \)

### Step 3: Find a Common Denominator
To compare these two fractions, we need to have a common denominator. The denominators are \( 17 \) and \( 7 \). The least common multiple (LCM) of \( 17 \) and \( 7 \) is \( 119 \).

### Step 4: Convert Both Fractions
Now we will convert both fractions to have the same denominator of \( 119 \).

- For \( -\frac{12}{17} \):
\[ -\frac{12}{17} = -\frac{12 \times 7}{17 \times 7} = -\frac{84}{119} \]

- For \( \frac{-6}{7} \):
\[ \frac{-6}{7} = \frac{-6 \times 17}{7 \times 17} = -\frac{102}{119} \]

### Step 5: Compare the Two Fractions
Now we compare \( -\frac{84}{119} \) and \( -\frac{102}{119} \).

Since both fractions have the same denominator, we can compare the numerators:
- The numerators are \( -84 \) and \( -102 \).

### Step 6: Determine Which is Greater
In the case of negative numbers, a smaller number (more negative) is actually greater.
- Since \( -84 > -102 \), we have:
\[ -\frac{84}{119} > -\frac{102}{119} \]

### Conclusion
Thus, we can conclude that:
\[ \frac{12}{-17} > \frac{-6}{7} \]