Question
Identify the graph of the solution set for the inequality x−2y<2
.(1 point)
Responses
a
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
b.
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 1.5 comma negative 5 right parenthesis, passes through the two plotted points, and ends at left parenthesis 3.5 comma 5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 1.5 comma negative 5 right parenthesis, passes through the two plotted points, and ends at left parenthesis 3.5 comma 5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
c.
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area above the dashed line is shaded. All points are unlabeled.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area above the dashed line is shaded. All points are unlabeled.
d.
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 1.5 comma negative 5 right parenthesis, passes through the two plotted points, and ends at left parenthesis 3.5 comma 5 right parenthesis. The area above the dashed line is shaded. All points are unlabeled.
.(1 point)
Responses
a
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
b.
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 1.5 comma negative 5 right parenthesis, passes through the two plotted points, and ends at left parenthesis 3.5 comma 5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 1.5 comma negative 5 right parenthesis, passes through the two plotted points, and ends at left parenthesis 3.5 comma 5 right parenthesis. The area below the dashed line is shaded. All points are unlabeled.
c.
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area above the dashed line is shaded. All points are unlabeled.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 1 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 5 comma negative 3.5 right parenthesis, passes through the two points, and ends at left parenthesis 5 comma 1.5 right parenthesis. The area above the dashed line is shaded. All points are unlabeled.
d.
A coordinate plane shows four quadrants with both the axes ranging from negative 5 to 5 in increments of 1. Two points are plotted and a dashed line passes through them. The two points are plotted at coordinates left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 0 right parenthesis. The dashed line starts at left parenthesis negative 1.5 comma negative 5 right parenthesis, passes through the two plotted points, and ends at left parenthesis 3.5 comma 5 right parenthesis. The area above the dashed line is shaded. All points are unlabeled.
Answers
GPT-4o mini
To solve the inequality \( x - 2y < 2 \), we start by rewriting it in slope-intercept form. First, we isolate \( y \):
\[
x - 2y < 2
\]
Subtract \( x \) from both sides:
\[
-2y < -x + 2
\]
Now, divide everything by -2 and remember to flip the inequality sign:
\[
y > \frac{1}{2}x - 1
\]
This means we are looking for the region above the line \( y = \frac{1}{2}x - 1 \).
Next, we can find some points on the line to graph it. For example:
- When \( x = 0 \):
\[
y = \frac{1}{2}(0) - 1 = -1 \quad \text{(point: } (0, -1))
\]
- When \( x = 2 \):
\[
y = \frac{1}{2}(2) - 1 = 0 \quad \text{(point: } (2, 0))
\]
From this, we know our line passes through the points (0, -1) and (2, 0).
Now, since the inequality is "<" (not "≤"), we represent this line as a dashed line. The region we are concerned with is above this line.
Examining the options given in the responses, we see:
- **Option a**: Shows a dashed line through (0, -1) and (2, 0) with shading below the line (incorrect).
- **Option b**: Shows different points and a line (incorrect).
- **Option c**: Shows a dashed line through (0, -1) and (2, 0) but shades above the line (correct).
- **Option d**: Also shows different points with a shaded area above but fits the wrong points for this inequality (incorrect).
The correct response describing the graph of the solution set for the inequality \( x - 2y < 2 \) is:
**c**: A coordinate plane showing a dashed line passing through the points (0, -1) and (2, 0) with the area above the line shaded.
\[
x - 2y < 2
\]
Subtract \( x \) from both sides:
\[
-2y < -x + 2
\]
Now, divide everything by -2 and remember to flip the inequality sign:
\[
y > \frac{1}{2}x - 1
\]
This means we are looking for the region above the line \( y = \frac{1}{2}x - 1 \).
Next, we can find some points on the line to graph it. For example:
- When \( x = 0 \):
\[
y = \frac{1}{2}(0) - 1 = -1 \quad \text{(point: } (0, -1))
\]
- When \( x = 2 \):
\[
y = \frac{1}{2}(2) - 1 = 0 \quad \text{(point: } (2, 0))
\]
From this, we know our line passes through the points (0, -1) and (2, 0).
Now, since the inequality is "<" (not "≤"), we represent this line as a dashed line. The region we are concerned with is above this line.
Examining the options given in the responses, we see:
- **Option a**: Shows a dashed line through (0, -1) and (2, 0) with shading below the line (incorrect).
- **Option b**: Shows different points and a line (incorrect).
- **Option c**: Shows a dashed line through (0, -1) and (2, 0) but shades above the line (correct).
- **Option d**: Also shows different points with a shaded area above but fits the wrong points for this inequality (incorrect).
The correct response describing the graph of the solution set for the inequality \( x - 2y < 2 \) is:
**c**: A coordinate plane showing a dashed line passing through the points (0, -1) and (2, 0) with the area above the line shaded.