Question

x - 6y > 18

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(x - 6y>18\). First, we want to solve for \(y\). Subtract \(x\) from both sides: \(- 6y>-x + 18\). Then, divide each term by \(-6\). When we divide an inequality by a negative number, the direction of the inequality sign changes. So we get \(y<\frac{1}{6}x - 3\).

Step2: Graph the boundary line

The boundary line for the inequality \(y<\frac{1}{6}x - 3\) is the line \(y=\frac{1}{6}x - 3\). Since the inequality is strict (\(y<\frac{1}{6}x - 3\), not \(y\leq\frac{1}{6}x - 3\)), we draw a dashed line. To find two points on the line \(y = \frac{1}{6}x-3\), we can use the \(x\) - intercept and \(y\) - intercept.

• For the \(y\) - intercept, set \(x = 0\). Then \(y=\frac{1}{6}(0)-3=-3\). So the point \((0,-3)\) is on the line.
• For the \(x\) - intercept, set \(y = 0\). Then \(0=\frac{1}{6}x-3\). Add 3 to both sides: \(\frac{1}{6}x=3\). Multiply both sides by 6: \(x = 18\). So the point \((18,0)\) is on the line.

Step3: Determine the region to shade

We test a point not on the line. A common test point is the origin \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality \(y<\frac{1}{6}x - 3\). We get \(0<\frac{1}{6}(0)-3\), which simplifies to \(0 < - 3\). This is false. So we shade the region that does not contain the origin. In other words, we shade the region below the dashed line \(y=\frac{1}{6}x - 3\).

To graph \(x - 6y>18\):

1. Draw the dashed line \(y=\frac{1}{6}x - 3\) (using the points \((0, - 3)\) and \((18,0)\)).
2. Shade the region below the dashed line (since the test point \((0,0)\) does not satisfy the inequality, we shade the opposite side of the line from the origin).

Answer:

To graph \(x - 6y>18\):

1. Rewrite as \(y<\frac{1}{6}x - 3\).
2. Draw a dashed line \(y=\frac{1}{6}x - 3\) (through \((0,-3)\) and \((18,0)\)).
3. Shade the region below the dashed line.