Question

graph the inequality on the axes below.
x - 2y < -4

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(x - 2y<-4\). First, we want to solve for \(y\).
Subtract \(x\) from both sides: \(- 2y<-x - 4\).
Then, divide each term by \(-2\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So we get \(y>\frac{1}{2}x + 2\).

Step2: Graph the boundary line

The boundary line for the inequality \(y>\frac{1}{2}x + 2\) is the line \(y=\frac{1}{2}x + 2\). Since the inequality is \(y>\frac{1}{2}x + 2\) (not \(y\geq\frac{1}{2}x + 2\)), the boundary line should be a dashed line.
To graph the line \(y=\frac{1}{2}x + 2\), we can find the \(y\) - intercept and the slope. The \(y\) - intercept \(b = 2\), so the line passes through the point \((0,2)\). The slope \(m=\frac{1}{2}\), which means from the point \((0,2)\), we can go up 1 unit and to the right 2 units to get another point, for example, \((2,3)\). We draw a dashed line through these points.

Step3: Shade the region

Since the inequality is \(y>\frac{1}{2}x + 2\), we need to shade the region above the dashed line \(y = \frac{1}{2}x+2\). We can test a point that is not on the line, for example, the origin \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>\frac{1}{2}x + 2\): \(0>\frac{1}{2}(0)+2\), which simplifies to \(0 > 2\), which is false. So the origin is not in the solution region. Therefore, we shade the region above the dashed line.

Answer:

The graph has a dashed line \(y=\frac{1}{2}x + 2\) (passing through \((0,2)\) and \((2,3)\) etc.) and the region above the line is shaded.