Which description of the graph of the linear inequality y ≥ 7x – 4 is correct?

Question

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.
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.
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.
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.

Answer 1

To analyze the linear inequality \( y \geq 7x - 4 \), let's break down its components:

1. Line Representation: The related equation is \( y = 7x - 4 \).

   • Y-Intercept: This is the point where the line crosses the y-axis, which occurs when \( x = 0 \). Plugging in \( x = 0 \): \[ y = 7(0) - 4 = -4 \] So the y-intercept is \( -4 \).

   • Slope: The slope of the line is the coefficient of \( x \), which is \( 7 \).

2. Solid vs. Dashed Line:

   • The inequality sign \( \geq \) indicates that the line should be solid (because points on the line itself satisfy the inequality).

3. Shading Direction:

   • Since the inequality is \( y \geq 7x - 4 \), we will shade above the line. This represents all the points where the \( y \)-value is greater than or equal to the linear expression on the right.

With this information established, the correct description of the graph is:

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.