To determine which inequality is correct, let us analyze each given option relative to the statement "[tex]$-\frac{1}{2}$[/tex] is a minimum of the product of a number and [tex]$-\frac{5}{6}$[/tex]":
### Step-by-Step Analysis:
1. Inequality Option 1:
[tex]\[
-\frac{5}{6} \leq -\frac{1}{2} w
\][/tex]
This inequality states that [tex]\(-\frac{5}{6}\)[/tex] is less than or equal to [tex]\(-\frac{1}{2} w\)[/tex].
2. Inequality Option 2:
[tex]\[
-\frac{5}{6} \geq -\frac{1}{2} w
\][/tex]
This inequality states that [tex]\(-\frac{5}{6}\)[/tex] is greater than or equal to [tex]\(-\frac{1}{2} w\)[/tex].
3. Inequality Option 3:
[tex]\[
-\frac{1}{2} \geq -\frac{5}{6} w
\][/tex]
This inequality states that [tex]\(-\frac{1}{2}\)[/tex] is greater than or equal to [tex]\(-\frac{5}{6} w\)[/tex].
4. Inequality Option 4:
[tex]\[
-\frac{1}{2} \leq -\frac{5}{6} w
\][/tex]
This inequality states that [tex]\(-\frac{1}{2}\)[/tex] is less than or equal to [tex]\(-\frac{5}{6} w\)[/tex].
### Conclusion:
By reviewing the numerical value in each case, we notice that:
- The correct relationship between [tex]\(-\frac{1}{2}\)[/tex] and [tex]\(-\frac{5}{6}\)[/tex] needs to be considered to identify the correct inequality.
We find that:
[tex]\[
-\frac{5}{6} \approx -0.8333 \quad \text{and} \quad -\frac{1}{2} \approx -0.5
\][/tex]
Given that [tex]\(-0.5\)[/tex] ([tex]\(-\frac{1}{2}\)[/tex]) is greater than [tex]\(-0.8333\)[/tex] ([tex]\(-\frac{5}{6}\)[/tex]), we can conclude the correct inequality is:
[tex]\[
-\frac{5}{6} \geq -\frac{1}{2} w
\][/tex]
Thus, the correct inequality is identified as:
[tex]\[
-\frac{5}{6} \geq -\frac{1}{2} w
\][/tex]
Which corresponds to Option 2.
