Question

Which of the following pints is a solution to the linear inequality y>\frac{1}{9}x-\frac{2}{9}
(0, -2/9)
(0, -4/9)
(0, -1/9)
(0, -1/3)

Answer 1

To determine which points are solutions to the inequality \( y > \frac{1}{9}x - \frac{2}{9} \), we can substitute the x-value of each point into the inequality and check if the corresponding y-value satisfies the inequality.

Let's analyze each point one by one:

1. **Point (0, -2/9)**:
- Substitute \( x = 0 \):
\[
y > \frac{1}{9}(0) - \frac{2}{9} \implies y > -\frac{2}{9}
\]
- Check \( y = -\frac{2}{9} \):
\[
-\frac{2}{9} \not> -\frac{2}{9} \quad \text{(false)}
\]
- Thus, (0, -2/9) is **not a solution**.

2. **Point (0, -4/9)**:
- Substitute \( x = 0 \):
\[
y > -\frac{2}{9}
\]
- Check \( y = -\frac{4}{9} \):
\[
-\frac{4}{9} \not> -\frac{2}{9} \quad \text{(false)}
\]
- Thus, (0, -4/9) is **not a solution**.

3. **Point (0, -1/9)**:
- Substitute \( x = 0 \):
\[
y > -\frac{2}{9}
\]
- Check \( y = -\frac{1}{9} \):
\[
-\frac{1}{9} > -\frac{2}{9} \quad \text{(true)}
\]
- Thus, (0, -1/9) **is a solution**.

4. **Point (0, -1/3)**:
- Substitute \( x = 0 \):
\[
y > -\frac{2}{9}
\]
- Check \( y = -\frac{1}{3} \) (which is \(-\frac{3}{9}\)):
\[
-\frac{1}{3} \not> -\frac{2}{9} \quad \text{(false)}
\]
- Thus, (0, -1/3) is **not a solution**.

In conclusion, the only point that is a solution to the inequality \( y > \frac{1}{9}x - \frac{2}{9} \) is (0, -1/9).