Question

On a coordinate plane, 2 straight lines are shown. The first solid line has a positive slope and goes through (0, negative 2) and (2, 0). Everything to the left of the line is shaded. The second dashed line has a negative slope and goes through (0, 2) and (4, 0). Everything below and to the left of the line is shaded.
Which system of linear inequalities is represented by the graph?

y > x – 2 and x – 2y < 4
y > x + 2 and x + 2y < 4
y > x – 2 and x + 2y < 4
y > x – 2 and x + 2y < –4

Answer 1

To determine the system of inequalities represented by the graph, we first analyze the two lines given:

1. **First Line (Solid Line)**: This line has a positive slope and passes through the points (0, -2) and (2, 0).
- The equation can be determined using the slope-intercept form \(y = mx + b\).
- The slope (m) is calculated as \((0 - (-2)) / (2 - 0) = 1\).
- The line can be described by the equation:
\[
y = x - 2
\]
- Since the area to the left of this line is shaded, this translates to the inequality:
\[
y > x - 2
\]

2. **Second Line (Dashed Line)**: This line has a negative slope and passes through (0, 2) and (4, 0).
- The slope here is \((0 - 2) / (4 - 0) = -1/2\).
- The equation of the line can be derived as:
\[
y = -\frac{1}{2}x + 2
\]
- Rearranging it gives:
\[
x + 2y < 4
\]
- Because the area below and to the left of this line is shaded, this translates to the inequality:
\[
x + 2y < 4
\]

Combining these two inequalities, we have:
- From the first line: \(y > x - 2\)
- From the second line: \(x + 2y < 4\)

Thus, the system of linear inequalities represented by the graph is:
\[
y > x - 2 \quad \text{and} \quad x + 2y < 4
\]

The correct choice from the given options is:
\[
\boxed{y > x - 2 \text{ and } x + 2y < 4}
\]