Question

Use the image to answer the 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.**