Question

Question was provided via image upload.

Explanation:

Identify the inequality and boundary line

The linear inequality shown in the image is:

$$ y \le \frac{2}{3}x - 6 $$

The corresponding boundary line is:

$$ y = \frac{2}{3}x - 6 $$

Determine the line style and points

Since the inequality symbol is \(\le\) (less than or equal to), the boundary line is solid.
We find two points on the line:

• When \(x = 0\):

$$ y = -6 \implies (0, -6) $$

• When \(x = 9\):

$$ y = \frac{2}{3}(9) - 6 = 0 \implies (9, 0) $$

Determine the shaded region

Since the inequality is \(y \le \frac{2}{3}x - 6\), we shade the region below the solid boundary line.
Testing the origin \((0,0)\):

$$ 0 \le \frac{2}{3}(0) - 6 \implies 0 \le -6 \quad (\text{False}) $$

Thus, the region containing \((0,0)\) is not shaded; the region below the line is shaded.

Answer:

To graph the linear inequality \(y \le \frac{2}{3}x - 6\):

1. Draw the boundary line: Plot a solid line passing through the y-intercept \((0, -6)\) and the x-intercept \((9, 0)\).
2. Shade the region: Shade the region below the boundary line.