Question

Which of the following points is a solution to the linear inequality: y > 1/9 x - 2/9 (0, - 4/9), (0, - 1/9), (0, - 2/9), (0, - 1/3)

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 can substitute each point into the inequality and check if the inequality holds true.

The inequality in question is:
\[
y > \frac{1}{9}x - \frac{2}{9}
\]

Let's evaluate each point:

1. **Point (0, -4/9)**:
\[
y = -\frac{4}{9}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{4}{9} > -\frac{2}{9}
\]
This is **false**.

2. **Point (0, -1/9)**:
\[
y = -\frac{1}{9}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{1}{9} > -\frac{2}{9}
\]
This is **true**.

3. **Point (0, -2/9)**:
\[
y = -\frac{2}{9}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{2}{9} > -\frac{2}{9}
\]
This is **false** (because it’s not greater).

4. **Point (0, -1/3)**:
\[
y = -\frac{1}{3}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{1}{3} > -\frac{2}{9}
\]
To compare -1/3 and -2/9, convert -1/3 to a fraction with common denominator 9:
\[
-\frac{1}{3} = -\frac{3}{9}
\]
Then we check:
\[
-\frac{3}{9} > -\frac{2}{9}
\]
This is **false**.

The only point that satisfies the inequality is **(0, -1/9)**.