Question

list the expressions x, x-2, x^2, and 2x-6 in order from least to greatest for all x < 0

Answer 1

To compare the expressions \( x \), \( x - 2 \), \( x^2 \), and \( 2x - 6 \) for all \( x < 0 \), we can analyze how each expression behaves as \( x \) takes negative values.

1. **Analyzing the expressions**:
- \( x \): Since \( x < 0 \), this will always be negative.
- \( x - 2 \): This is also negative and will be less than \( x \) since subtracting 2 makes it smaller.
- \( x^2 \): As \( x \) is negative, \( x^2 \) will be positive (since squaring a negative number yields a positive result).
- \( 2x - 6 \): Since \( 2x \) is negative (as \( x < 0 \)), and subtracting 6 makes it even more negative. Thus, this expression will also be negative.

2. **Finding relationships between the negative expressions**:
- **Comparing \( x \) and \( x - 2 \)**:
\[
x - 2 < x \quad \text{(always true)}
\]
- **Comparing \( x \) and \( 2x - 6 \)**:
\[
2x - 6 < x \implies 2x < x + 6 \implies x < 6 \quad \text{(true for all } x < 0\text{)}
\]
- **Comparing \( x - 2 \) and \( 2x - 6 \)**:
\[
2x - 6 < x - 2 \implies 2x < x + 4 \implies x < 4 \quad \text{(also true for all } x < 0\text{)}
\]

3. **Finding the relationship between \( 2x - 6 \) and \( x^2 \)**:
- We need to find when \( 2x - 6 < x^2 \):
\[
x^2 - 2x + 6 > 0
\]
The discriminant of the quadratic \( x^2 - 2x + 6 \) is:
\[
(-2)^2 - 4(1)(6) = 4 - 24 = -20 \quad \text{(negative, so always positive)}
\]
Hence, \( 2x - 6 < x^2 \) for all \( x \).

4. **Summarizing the results**:
- We have determined:
- \( x - 2 < x \)
- \( 2x - 6 < x \)
- \( 2x - 6 < x - 2 \)
- \( 2x - 6 < x^2 \)
- \( x < x^2 \) when \( x < 0 \)

5. **Order the expressions**:
Thus, we can conclude for all \( x < 0\):
\[
2x - 6 < x - 2 < x < x^2
\]

The final order from least to greatest for all \( x < 0 \) is:
\[
2x - 6, \ x - 2, \ x, \ x^2
\]