Use the image to answer the question.

Question

A coordinate plane shows a graphed dashed line. The x-axis and y-axis both range from negative 3 to 3 in one unit increments. No points are plotted. The graphed line has a y-intercept at coordinates left parenthesis 0 comma negative 2.5 right parenthesis. The x-intercept is approximately at coordinates left parenthesis 1.5 comma 0 right parenthesis. The area above the dashed line is shaded.

Determine the solution set for the given graph of an inequality in two variables.

Answer 1

To determine the solution set for the inequality represented by the dashed line described in the graph, we can analyze the information provided:

1. Y-intercept: The line intersects the y-axis at \( (0, -2.5) \).
2. X-intercept: The line intersects the x-axis at approximately \( (1.5, 0) \).
3. Shading: The area above the dashed line is shaded, indicating that this area represents the solutions to the inequality.

With these features, we can deduce the equation of the line. The line's slope \( m \) can be calculated using the y-intercept and x-intercept:

• The change in \( y \) (from \( -2.5 \) to \( 0 \)) is \( 2.5 \).
• The change in \( x \) (from \( 0 \) to \( 1.5 \)) is \( 1.5 \).

Calculating the slope \( m \):

\[ m = \frac{\Delta y}{\Delta x} = \frac{2.5}{1.5} \approx \frac{5}{3} \]

Thus, we can express the line's equation in point-slope form:

\[ y - (-2.5) = \frac{5}{3}(x - 0) \]

Simplifying this gives:

\[ y + 2.5 = \frac{5}{3}x \] \[ y = \frac{5}{3}x - 2.5 \]

Since the line is dashed, it indicates that the inequality is strict (not including the line itself). As the shaded area is above the line, the solution set is represented by:

\[ y > \frac{5}{3}x - 2.5 \]

Thus, the solution set for the given graph of the inequality in two variables is:

\[ {(x, y) | y > \frac{5}{3}x - 2.5} \]