Which description of the graph of the linear inequality $y>3 x-8$ is correct?
A. The graph will be a dashed line with a $y$-intercept of negative eight and a slope of three. The graph will be shaded below the line.
B. The graph will be a solid line with a $y$-intercept of three and a slope of negative eight. The graph will be shaded above the line.
C. The graph will be a solid line with a $y$-intercept of three and a slope of negative eight. The graph will be shaded below the line.
D. The graph will be a dashed line with a $y$-intercept of negative eight and a slope of three. The graph will be shaded above the line.
Answer (1)
The graph is a dashed line because the inequality is strict ( "> > ).
The y-intercept is -8 and the slope is 3, derived from the slope-intercept form of the inequality.
The region above the line is shaded because the inequality is 3x - 8"> y > 3 x − 8 .
The correct description is: The graph will be a dashed line with a y -intercept of negative eight and a slope of three. The graph will be shaded above the line. \boxed{The graph will be a dashed line with a y$-intercept of negative eight and a slope of three. The graph will be shaded above the line.}
Explanation
Analyzing the Inequality The given linear inequality is 3x - 8"> y > 3 x − 8 . We need to identify the characteristics of its graph: whether the line is solid or dashed, the y-intercept, the slope, and the shaded region.
Identifying Slope and Y-intercept The inequality is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In our case, m = 3 and b = − 8 . So, the slope is 3 and the y-intercept is -8.
Determining the Line Type Since the inequality is 3x - 8"> y > 3 x − 8 , and not = 3x - 8"> y " >= 3 x − 8 , the line is dashed, indicating that the points on the line are not included in the solution.
Determining the Shaded Region The inequality is 3x - 8"> y > 3 x − 8 , which means we are looking for all the points where the y-value is greater than 3 x − 8 . This corresponds to the region above the line.
Conclusion Therefore, the graph will be a dashed line with a y-intercept of -8 and a slope of 3, and the graph will be shaded above the line.
Examples
Linear inequalities are used in various real-world scenarios, such as determining the feasible region in linear programming problems. For example, a company might use a linear inequality to represent the constraint on the number of hours available for production, where y represents the number of units of one product and x represents the number of units of another product. The inequality 3x - 8"> y > 3 x − 8 could represent a constraint where the production of product y must be greater than three times the production of product x minus 8, to ensure profitability or meet demand.
