Use the image to answer the question.

Question

A coordinate plane shows a graphed line. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. No points are plotted. The graphed line has a y-intercept at coordinates left parenthesis 0 comma 1 right parenthesis. The x-intercept is at coordinates left parenthesis 1 comma 0 right parenthesis. The area above the solid line is shaded.

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

(1 point)
Responses

x+y>1
x plus y greater than 1

−x+y≥1
negative x plus y greater-than-or-equal-to 1

x+y≥1
x plus y greater-than-or-equal-to 1

y≥x+1

Answer 1

To determine the correct inequality that corresponds to the given graph, we can analyze the information provided:

1. The line has a y-intercept at (0, 1), which means the line crosses the y-axis at 1.
2. The line has an x-intercept at (1, 0), which means it crosses the x-axis at 1.
3. The area above the solid line is shaded.

The equation of the line can be derived from the intercepts. The two intercepts can be used to find the slope:

• The slope is calculated as:

\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{1 - 0} = -1 \]

Using the slope-intercept form \(y = mx + b\):

• Here, \(m = -1\) and \(b = 1\).
• Thus, the equation of the line is:

\[ y = -x + 1 \]

Rearranging this equation:

\[ x + y = 1 \]

Since the area above the line is shaded, this indicates that we are looking for the region where \(y\) is greater than the line. Therefore, we have:

\[ y > -x + 1 \quad \text{or} \quad x + y > 1 \]

Now, looking at the possible responses:

1. \( x+y > 1 \)
2. \( -x+y \geq 1 \)
3. \( x+y \geq 1 \)
4. \( y \geq x+1 \)

The inequality that represents the area above the line accurately is:

x + y > 1.

Thus, the correct response is:

x+y>1.