On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 3, 1) and (0, 3). Everything to the left of the line is shaded.

Question

Which linear inequality is represented by the graph?

y < Two-thirdsx + 3
y > Three-halvesx + 3
y > Two-thirdsx + 3
y < Three-halvesx + 3

Answer 1

To determine the correct linear inequality represented by the graph, we first need to find the equation of the dashed line.

The points given are (-3, 1) and (0, 3). We can find the slope (\(m\)) of the line using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{0 - (-3)} = \frac{2}{3} \]

Now that we have the slope, we can use the point-slope form of a linear equation. Using the point (0, 3), which is also the y-intercept (\(b\)), we can write the equation of the line:

\[ y = \frac{2}{3}x + 3 \]

Since the line is dashed, it indicates that the inequality is strict (< or >). From the description, the region shaded is to the left of this line. Therefore, we want the inequality where \(y\) is less than the line equation.

Thus, the inequality is:

\[ y < \frac{2}{3}x + 3 \]

Looking at the options given, the correct inequality that represents the graph is:

\[ y < \frac{2}{3}x + 3 \]

So, the answer is:

y < Two-thirdsx + 3.