Question

graph the following inequality.
$4x + y > -4$
use the graphing tool to graph the inequality.
click to enlarge graph

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(4x + y>-4\). To get it in the form \(y = mx + b\) (slope - intercept form), we solve for \(y\). Subtract \(4x\) from both sides of the inequality:
\(y>-4x - 4\)

Step2: Analyze the boundary line

The boundary line for the inequality \(y>-4x - 4\) is the line \(y=-4x - 4\). Since the inequality is \(y >-4x - 4\) (and not \(y\geq-4x - 4\)), the boundary line should be a dashed line. This is because the points on the line \(y =-4x-4\) do not satisfy the inequality \(y>-4x - 4\).

Step3: Determine the slope and y - intercept of the boundary line

For the line \(y=-4x - 4\), the slope \(m=-4\) and the y - intercept \(b = - 4\). To graph the line, we start at the y - intercept \((0,-4)\). Then, using the slope (rise over run), since the slope is \(-4=\frac{-4}{1}\), from the point \((0,-4)\), we can go down 4 units and right 1 unit (or up 4 units and left 1 unit) to find other points on the line. For example, from \((0,-4)\), moving down 4 and right 1 gives us the point \((1,-8)\), and moving up 4 and left 1 gives us the point \((-1,0)\).

Step4: Shade the region

To determine which side of the line to shade, we can use a test point. A common test point is the origin \((0,0)\) (as long as it is not on the boundary line). Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>-4x - 4\):
\(0>-4(0)-4\)
\(0> - 4\)
This statement is true. So we shade the region that contains the origin \((0,0)\).

Answer:

To graph \(4x + y>-4\):

1. Rewrite as \(y>-4x - 4\).
2. Graph the dashed line \(y=-4x - 4\) (slope \(-4\), y - intercept \(-4\)).
3. Shade the region above the dashed line (the region containing \((0,0)\)).