If a is an integer less than `-1`. Which of the following orders the expressions `|a|, -a^(2), and -1/a` from least value to greatest value?

Options

A. `-1/a lt -a^(2) lt |a|`
B. `-1/a lt |a| lt -a^(2)`
C. `|a| lt -1/a lt -a^(2)`
D. `-a^(2) lt -1/a lt |a| `

Answer

To solve the problem, we need to compare the three expressions: \(|a|\), \(-a^2\), and \(-\frac{1}{a}\) given that \(a\) is an integer less than \(-1\).

### Step-by-Step Solution:

1. **Understanding the Absolute Value**:
Since \(a < -1\), the absolute value \(|a|\) is equal to \(-a\). Therefore, we can rewrite \(|a|\) as:
\[ |a| = -a \]
Since \(a\) is negative, \(-a\) will be a positive number greater than \(1\).

**Hint**: Remember that the absolute value of a negative number is its positive counterpart.

2. **Analyzing \(-a^2\)**:
Since \(a < -1\), squaring \(a\) gives:
\[ a^2 > 1 \]
Therefore, \(-a^2\) will be:
\[ -a^2 < -1 \]
This means \(-a^2\) is a negative number less than \(-1\).

**Hint**: Squaring any number less than \(-1\) results in a positive number greater than \(1\).

3. **Analyzing \(-\frac{1}{a}\)**:
Since \(a < -1\), the reciprocal \(-\frac{1}{a}\) will be:
\[ -\frac{1}{a} > 1 \]
This is because dividing by a negative number less than \(-1\) results in a positive number greater than \(1\).

**Hint**: The reciprocal of a negative number less than \(-1\) is positive and greater than \(1\).

4. **Ordering the Expressions**:
Now we have:
- \(-a^2 < -1\) (which is negative)
- \(-\frac{1}{a} > 1\) (which is positive)
- \(|a| = -a > 1\) (which is also positive)

Since \(-a^2\) is negative and both \(-\frac{1}{a}\) and \(|a|\) are positive, we can order them as follows:
\[ -a^2 < -\frac{1}{a} \quad \text{and} \quad -\frac{1}{a} < |a| \]

Therefore, the complete ordering from least to greatest is:
\[ -a^2 < -\frac{1}{a} < |a| \]

5. **Final Conclusion**:
The correct ordering of the expressions from least value to greatest value is:
\[ -a^2 < -\frac{1}{a} < |a| \]
Thus, the answer is option 4.