Let's analyze the linear inequality [tex]\( y \geq 7x - 4 \)[/tex]:
1. Form of the Inequality:
- The inequality is given in the form [tex]\( y \geq mx + b \)[/tex].
- Here, [tex]\( m = 7 \)[/tex] (this is the slope) and [tex]\( b = -4 \)[/tex] (this is the y-intercept).
2. Solid vs. Dashed Line:
- Because the inequality sign is [tex]\( \geq \)[/tex] (greater than or equal to), the line should be solid. Solid lines indicate that the points on the line satisfy the inequality.
3. Slope and y-Intercept:
- The slope [tex]\( m \)[/tex] is 7, which indicates how steep the line is.
- The y-intercept [tex]\( b \)[/tex] is -4, which is the point where the line crosses the y-axis.
4. Direction of Shading:
- Since the inequality is [tex]\( y \geq 7x - 4 \)[/tex], we shade above the line. For [tex]\( \geq \)[/tex] signs, the region above the line represents all the points [tex]\((x, y)\)[/tex] that satisfy the inequality.
Given these points:
- The line is solid because it includes the equality.
- The line has a y-intercept at -4.
- The line has a slope of 7.
- The shading is above the line.
From the given options:
1. The graph will be a dashed line with a [tex]\( y \)[/tex]-intercept of negative four and a slope of seven. The graph will be shaded below the line.
- Incorrect because the line should be solid and the shading should be above the line.
2. 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.
- Correct because this matches all of our observations.
3. The graph will be a solid line with a [tex]\( y \)[/tex]-intercept of seven and a slope of negative four. The graph will be shaded below the line.
- Incorrect because the y-intercept and slope are incorrect, and the shading direction is wrong.
4. The graph will be a dashed line with a [tex]\( y \)[/tex]-intercept of seven and a slope of negative four. The graph will be shaded above the line.
- Incorrect because the line should be solid, and the y-intercept and slope are incorrect.
Based on this analysis, the correct description 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.
