Answer :
To solve the problem of describing the graph of the linear inequality \( y \geq 7x - 4 \), we must break down the components and characteristics of the inequality step-by-step.
1. Identify the slope and intercept:
- The inequality \( y \geq 7x - 4 \) is in slope-intercept form \( y \geq mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Here, the slope \( m \) is \( 7 \) and the y-intercept \( b \) is \( -4 \).
2. Type of Line:
- The inequality symbol is \( \geq \), indicating that the boundary line is included in the solution set. Therefore, the line should be solid.
3. Shading:
- The inequality \( y \geq 7x - 4 \) means the region above the line \( y = 7x - 4 \) is included in the solution set.
- This means we will shade the region above the line.
Given these steps:
- The graph will have a solid line.
- The y-intercept of the line is at \(-4\).
- The slope of the line is \(7\).
- The graph should be shaded above the line.
Let's scrutinize the choices:
1. "The graph will be a dashed line with a \( y \)-intercept of negative four and a slope of seven. The graph will be shaded below the line."
- Incorrect: The line should be solid, not dashed, and the shading should be above, not below.
2. "The graph will be a solid line with a \( y \)-intercept of negative four and a slope of seven. The graph will be shaded above the line."
- Correct: This accurately describes a solid line with the correct y-intercept and slope, and the shading above the line.
3. "The graph will be a solid line with a \( y \)-intercept of seven and a slope of negative four. The graph will be shaded below the line."
- Incorrect: The y-intercept and slope are wrong, and the shading should be above, not below.
4. "The graph will be a dashed line with a \( y \)-intercept of seven and a slope of negative four. The graph will be shaded above the line."
- Incorrect: Both the y-intercept and slope are incorrect, and the line should be solid, not dashed.
Therefore, the correct description of the graph of the inequality \( y \geq 7x - 4 \) is:
- The graph will be a solid line with a [tex]\( y \)[/tex]-intercept of negative four and a slope of seven. The graph will be shaded above the line.
1. Identify the slope and intercept:
- The inequality \( y \geq 7x - 4 \) is in slope-intercept form \( y \geq mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Here, the slope \( m \) is \( 7 \) and the y-intercept \( b \) is \( -4 \).
2. Type of Line:
- The inequality symbol is \( \geq \), indicating that the boundary line is included in the solution set. Therefore, the line should be solid.
3. Shading:
- The inequality \( y \geq 7x - 4 \) means the region above the line \( y = 7x - 4 \) is included in the solution set.
- This means we will shade the region above the line.
Given these steps:
- The graph will have a solid line.
- The y-intercept of the line is at \(-4\).
- The slope of the line is \(7\).
- The graph should be shaded above the line.
Let's scrutinize the choices:
1. "The graph will be a dashed line with a \( y \)-intercept of negative four and a slope of seven. The graph will be shaded below the line."
- Incorrect: The line should be solid, not dashed, and the shading should be above, not below.
2. "The graph will be a solid line with a \( y \)-intercept of negative four and a slope of seven. The graph will be shaded above the line."
- Correct: This accurately describes a solid line with the correct y-intercept and slope, and the shading above the line.
3. "The graph will be a solid line with a \( y \)-intercept of seven and a slope of negative four. The graph will be shaded below the line."
- Incorrect: The y-intercept and slope are wrong, and the shading should be above, not below.
4. "The graph will be a dashed line with a \( y \)-intercept of seven and a slope of negative four. The graph will be shaded above the line."
- Incorrect: Both the y-intercept and slope are incorrect, and the line should be solid, not dashed.
Therefore, the correct description of the graph of the inequality \( y \geq 7x - 4 \) is:
- The graph will be a solid line with a [tex]\( y \)[/tex]-intercept of negative four and a slope of seven. The graph will be shaded above the line.
