Let's work through the inequality step by step to solve it:
1. Given inequality:
[tex]\[-\frac{1}{3} \geq -\frac{4}{5} y\][/tex]
2. Isolate [tex]\( y \)[/tex]:
We need to get [tex]\( y \)[/tex] by itself. To do this, we will divide both sides of the inequality by [tex]\(-\frac{4}{5}\)[/tex]. Remember, when we divide or multiply both sides of an inequality by a negative number, the direction of the inequality changes.
3. Divide by [tex]\(-\frac{4}{5}\)[/tex]:
[tex]\[
\frac{-\frac{1}{3}}{-\frac{4}{5}} \leq y
\][/tex]
4. Simplify the division:
[tex]\[
\frac{1}{3} \times \frac{5}{4} \leq y
\][/tex]
This simplifies to:
[tex]\[
\frac{1 \times 5}{3 \times 4} \leq y
\][/tex]
[tex]\[
\frac{5}{12} \leq y
\][/tex]
5. Write the inequality in standard form:
[tex]\[
y \geq \frac{5}{12}
\][/tex]
Now we will match this result with the given multiple choice answers:
- First Option: [tex]\(-\frac{1}{3}>-\frac{4}{5} y\)[/tex] where [tex]\(y \leq -\frac{4}{15}\)[/tex] (Doesn't match)
- Second Option: [tex]\(-\frac{1}{3} \geq -\frac{4}{5} y\)[/tex] where [tex]\(y \geq \frac{5}{12}\)[/tex] (Matches)
- Third Option: [tex]\(-\frac{4}{5} \geq -\frac{1}{3} y\)[/tex] where [tex]\(y \leq -\frac{5}{12}\)[/tex] (Doesn't match)
- Fourth Option: [tex]\(-\frac{4}{5} \leq -\frac{1}{3} y\)[/tex] where [tex]\(y \geq \frac{4}{15}\)[/tex] (Doesn't match)
The correct answer is:
[tex]\[
\boxed{2, \text{ "y} \geq \frac{\text{5}}{\text{12}}\text{"}}
\][/tex]
