Question
Enter an inequality that represents the graph in the box.
A coordinate grid from negative six to six along the x-axis in increments of 1 and from negative six to six along the y-axis in increments of 1. A graph of a line through begin ordered pair zero comma negative one end ordered pair and begin ordered pair one comma two end ordered pair. The region to the right and below the line is shaded.
A coordinate grid from negative six to six along the x-axis in increments of 1 and from negative six to six along the y-axis in increments of 1. A graph of a line through begin ordered pair zero comma negative one end ordered pair and begin ordered pair one comma two end ordered pair. The region to the right and below the line is shaded.
Answers
Answered by
GPT-4o mini
To represent the graph you've described, we first need to determine the equation of the line that passes through the points \((0, -1)\) and \((1, 2)\).
1. **Find the slope (m)**:
The slope \(m\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-1)}{1 - 0} = \frac{3}{1} = 3
\]
2. **Use the slope-intercept form of the line**:
We can use the point-slope form of the line equation, which is \(y - y_1 = m(x - x_1)\). Using the point \((0, -1)\):
\[
y - (-1) = 3(x - 0) \implies y + 1 = 3x \implies y = 3x - 1
\]
Thus, the equation of the line is:
\[
y = 3x - 1
\]
3. **Determine the inequality**:
Since the region shaded is to the right and below the line, we are interested in the area where \(y\) is less than the line. This leads us to the inequality:
\[
y < 3x - 1
\]
In conclusion, the inequality that represents the graph is:
\[
y < 3x - 1
\]
1. **Find the slope (m)**:
The slope \(m\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-1)}{1 - 0} = \frac{3}{1} = 3
\]
2. **Use the slope-intercept form of the line**:
We can use the point-slope form of the line equation, which is \(y - y_1 = m(x - x_1)\). Using the point \((0, -1)\):
\[
y - (-1) = 3(x - 0) \implies y + 1 = 3x \implies y = 3x - 1
\]
Thus, the equation of the line is:
\[
y = 3x - 1
\]
3. **Determine the inequality**:
Since the region shaded is to the right and below the line, we are interested in the area where \(y\) is less than the line. This leads us to the inequality:
\[
y < 3x - 1
\]
In conclusion, the inequality that represents the graph is:
\[
y < 3x - 1
\]