Question

which description of the graph of the linear inequality $y \geq 7x - 4$ is correct?
\bigcirc\\ 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.
\bigcirc\\ 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.
\bigcirc\\ 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.
\bigcirc\\ 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.

Explanation:

1. For a linear inequality in the form \(y \geq mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept):

• The line is solid because the inequality is "greater than or equal to" (if it were \(y>mx + b\), the line would be dashed).
• The slope \(m\) of the line \(y = 7x-4\) is \(7\) and the y - intercept \(b\) is \(- 4\) (since the equation is in slope - intercept form \(y=mx + b\)).
• For the inequality \(y\geq7x - 4\), we shade the region above the line because we are looking for all the points \((x,y)\) where \(y\) is greater than or equal to the value of \(7x - 4\).
• Let's analyze the other options:
• Option 1: The line should be solid (not dashed) for \(y\geq\) and we shade above (not below) the line.
• Option 3: The slope is \(7\) (not \(-4\)) and the y - intercept is \(-4\) (not \(7\)), also the shading direction is wrong.
• Option 4: The slope is \(7\) (not \(-4\)), the y - intercept is \(-4\) (not \(7\)), and the line should be solid (not dashed) and shaded above (not above with wrong slope and y - intercept).

Answer:

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.