Question

graph the following inequality.
x - 4y < 4
use the graphing tool to graph the inequality.
click to enlarge graph image

Explanation:

Step 1: Rewrite the inequality in slope - intercept form

We start with the inequality \(x - 4y<4\). We want to solve for \(y\) to get it in the form \(y = mx + b\) (slope - intercept form) where \(m\) is the slope and \(b\) is the y - intercept.
Subtract \(x\) from both sides: \(- 4y<-x + 4\)
Divide both sides by \(-4\). When we divide an inequality by a negative number, we must reverse the inequality sign. So we have \(y>\frac{1}{4}x - 1\)

Step 2: Graph the boundary line

The boundary line for the inequality \(y>\frac{1}{4}x - 1\) is the line \(y=\frac{1}{4}x - 1\). Since the inequality is \(y>\frac{1}{4}x - 1\) (and not \(y\geq\frac{1}{4}x - 1\)), the boundary line should be a dashed line.
To graph the line \(y=\frac{1}{4}x - 1\):

• The y - intercept \(b=- 1\), so the line passes through the point \((0,-1)\).
• The slope \(m = \frac{1}{4}\), which means from the point \((0,-1)\), we can go up 1 unit and then to the right 4 units to get another point on the line (for example, the point \((4,0)\)).

Step 3: Shade the region

Since the inequality is \(y>\frac{1}{4}x - 1\), we shade the region above the dashed line \(y = \frac{1}{4}x-1\). This is because for a point \((x,y)\) in the shaded region, the \(y\) - value will be greater than \(\frac{1}{4}x - 1\).

(Note: If we were to test a point, for example, the origin \((0,0)\): Substitute \(x = 0\) and \(y = 0\) into the original inequality \(x-4y<4\). We get \(0-4(0)=0<4\), which is true. And when we substitute into \(y>\frac{1}{4}x - 1\), we have \(0>\frac{1}{4}(0)-1=- 1\), which is also true. So the origin is in the solution region, and since the origin is above the line \(y=\frac{1}{4}x - 1\) (when \(x = 0\), \(y = 0\) and the line at \(x = 0\) has \(y=-1\)), we shade above the line.)

Answer:

To graph \(x - 4y<4\):

1. Rewrite as \(y>\frac{1}{4}x - 1\).
2. Graph the dashed line \(y=\frac{1}{4}x - 1\) (passing through \((0,-1)\) and \((4,0)\)).
3. Shade the region above the dashed line.