Question

Which of the following points is a solution to the linear inequality y>1/9x−2/9
𝑦
>
1
9
𝑥
−
2
9
?
Responses

(0,−19)
(
0
,
−
1
9
)
open paren 0 comma negative 1 ninth close paren - correct

(0,−49)
(
0
,
−
4
9
)
open paren 0 comma negative 4 ninths close paren - no response given

(0,−29)
(
0
,
−
2
9
)
open paren 0 comma negative 2 ninths close paren - no response given

(0,−13)

how to find the answer

Answer 1

To determine which of the given points is a solution to the linear inequality \( y > \frac{1}{9}x - \frac{2}{9} \), we will substitute the \( x \) value of each point into the inequality and check if the corresponding \( y \) value satisfies the inequality.

The inequality \( y > \frac{1}{9}x - \frac{2}{9} \) can be evaluated for each point \( (x, y) \).

### Point Evaluation

1. **Point (0, −19)**
Substitute \( x = 0 \):
\[
y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow y > -\frac{2}{9}
\]
Substitute \( y = -19 \):
\[
-19 > -\frac{2}{9} \quad \text{(False)}
\]

2. **Point (0, −1/9)**
Substitute \( x = 0 \):
\[
y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow y > -\frac{2}{9}
\]
Substitute \( y = -\frac{1}{9} \):
\[
-\frac{1}{9} > -\frac{2}{9} \quad \text{(True)}
\]

3. **Point (0, −4/9)**
Substitute \( x = 0 \):
\[
y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow y > -\frac{2}{9}
\]
Substitute \( y = -\frac{4}{9} \):
\[
-\frac{4}{9} > -\frac{2}{9} \quad \text{(False)}
\]

4. **Point (0, −2/9)**
Substitute \( x = 0 \):
\[
y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow y > -\frac{2}{9}
\]
Substitute \( y = -\frac{2}{9} \):
\[
-\frac{2}{9} > -\frac{2}{9} \quad \text{(False)}
\]

5. **Point (0, −1/3)**
Substitute \( x = 0 \):
\[
y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow y > -\frac{2}{9}
\]
Substitute \( y = -\frac{1}{3} = -\frac{3}{9} \):
\[
-\frac{3}{9} > -\frac{2}{9} \quad \text{(False)}
\]


### Summary of Results
- Point (0, −19) → False
- Point (0, −1/9) → True (This is a solution)
- Point (0, −4/9) → False
- Point (0, −2/9) → False
- Point (0, −1/3) → False

### Conclusion
The point **(0, −1/9)** is a solution to the linear inequality \( y > \frac{1}{9}x - \frac{2}{9} \).