Question
Graph the solution set of the inequality 2x−13y<−1
.(1 point)
Responses
a.
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the dashed line is shaded.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the dashed line is shaded.
b,
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in increments of 1. Two points are plotted and a solid line passes through them. The line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the solid line is shaded.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in increments of 1. Two points are plotted and a solid line passes through them. The line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the solid line is shaded.
c.
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area below the dashed line is shaded.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area below the dashed line is shaded.
d.
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a solid line passes through them. The line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area below the solid line is shaded.
.(1 point)
Responses
a.
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the dashed line is shaded.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the dashed line is shaded.
b,
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in increments of 1. Two points are plotted and a solid line passes through them. The line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the solid line is shaded.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in increments of 1. Two points are plotted and a solid line passes through them. The line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the solid line is shaded.
c.
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area below the dashed line is shaded.
Image with alt text: A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area below the dashed line is shaded.
d.
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a solid line passes through them. The line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area below the solid line is shaded.
Answers
Answered by
GPT-4o mini
To solve the inequality \(2x - 13y < -1\), we first rewrite it in slope-intercept form (y = mx + b).
1. Start with the inequality:
\[
2x - 13y < -1
\]
2. Isolate \(y\):
\[
-13y < -2x - 1
\]
3. Divide by \(-13\), remembering that this reverses the inequality sign:
\[
y > \frac{2}{13}x + \frac{1}{13}
\]
Now we have the line \(y = \frac{2}{13}x + \frac{1}{13}\).
### Step to Graph the Solution Set
- The line \(y = \frac{2}{13}x + \frac{1}{13}\) is where the inequality changes from '<' to '='. Since the original inequality is a strict inequality (<), we will use a dashed line to indicate that points on this line are **not** included in the solution set.
- The inequality \(y > \frac{2}{13}x + \frac{1}{13}\) means that we will shade the area **above** the line.
### Matching the Options
Given the descriptions of the graphs:
- **(a)** talks about a dashed line with the area above it shaded.
- **(b)** has a solid line and shades above, which is incorrect.
- **(c)** has a dashed line but shades below, which is incorrect.
- **(d)** has a solid line and shades below, also incorrect.
So, the correct response is:
**a.**
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the dashed line is shaded.
1. Start with the inequality:
\[
2x - 13y < -1
\]
2. Isolate \(y\):
\[
-13y < -2x - 1
\]
3. Divide by \(-13\), remembering that this reverses the inequality sign:
\[
y > \frac{2}{13}x + \frac{1}{13}
\]
Now we have the line \(y = \frac{2}{13}x + \frac{1}{13}\).
### Step to Graph the Solution Set
- The line \(y = \frac{2}{13}x + \frac{1}{13}\) is where the inequality changes from '<' to '='. Since the original inequality is a strict inequality (<), we will use a dashed line to indicate that points on this line are **not** included in the solution set.
- The inequality \(y > \frac{2}{13}x + \frac{1}{13}\) means that we will shade the area **above** the line.
### Matching the Options
Given the descriptions of the graphs:
- **(a)** talks about a dashed line with the area above it shaded.
- **(b)** has a solid line and shades above, which is incorrect.
- **(c)** has a dashed line but shades below, which is incorrect.
- **(d)** has a solid line and shades below, also incorrect.
So, the correct response is:
**a.**
A coordinate plane shows four quadrants with both the axes ranging from negative 4 to 4 in unit increments. Two points are plotted and a dashed line passes through them. The dashed line passes through two unlabeled points at left parenthesis minus 0.5 comma 0 right parenthesis and left parenthesis 0 comma 3 right parenthesis, extending beyond both points. The area above the dashed line is shaded.