Question

Which point is a solution to the inequality shown in this graph?
(-3, -1) to (3, 3)

Answer 1

To determine which point is a solution to the inequality represented by the line segment between the points (-3, -1) and (3, 3), we first need to find the equation of the line defined by these points.

1. **Calculate the slope (m)**:
The slope \( m \) between the points (-3, -1) and (3, 3) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3}
\]

2. **Use point-slope form** to find the equation of the line:
Using the point (-3, -1) and the slope \( \frac{2}{3} \):
\[
y - y_1 = m(x - x_1)
\]
Substituting in, we have:
\[
y - (-1) = \frac{2}{3}(x - (-3))
\]
Simplifying this gives:
\[
y + 1 = \frac{2}{3}(x + 3)
\]
Expanding this:
\[
y + 1 = \frac{2}{3}x + 2 \\
y = \frac{2}{3}x + 1
\]

3. **Determine the inequality**:
Since the graph shows a line segment from (-3, -1) to (3, 3), the solution to the inequality will likely either be above or below this line. Points below this line will satisfy \( y < \frac{2}{3}x + 1 \), while points above satisfy \( y > \frac{2}{3}x + 1 \).

4. **Testing points**:
Let's test a point below the line, like (0, 0):
\[
0 < \frac{2}{3}(0) + 1 \\
0 < 1 \quad \text{(True, so (0, 0) is a solution)}
\]

Now, let’s test (2, 2):
\[
2 > \frac{2}{3}(2) + 1 \\
2 > \frac{4}{3} + 1 \\
2 > \frac{7}{3} \quad \text{(False, so (2, 2) is not a solution)}
\]

Testing (3, 4):
\[
4 > \frac{2}{3}(3) + 1 \\
4 > 2 + 1 \\
4 > 3 \quad \text{(True, so (3, 4) is a solution)}
\]

To summarize, points below the line segment are solutions to the inequality. If you're looking for a specific solution that falls on or below the segment, you'd say:

- Any points such as (0, 0), (1, 1), (2, 2), etc., below the line are valid solutions.
- Ensure to check if you're looking for specific regions or boundary inclusion/exclusion to determine the exact inequality.

If you have other points you want to check, feel free to share!