Question

Sketch the graph of the inequality. $$9 x-3 y \geq 18$$

Short answer

The graph of the inequality \(9x - 3y \geq 18\) would be a line represented by \(y \leq 3x - 6\), with the area beneath the line shaded.

Step-by-step solution

Rearrange the inequality

Firstly, rearrange the given inequality \(9x - 3y \geq 18\) to the form \(y \leq mx + c\). Divide all terms by 3, it becomes \(3x - y \geq 6\), this can be rewritten as \(y \leq 3x - 6\). This form helps identify the slope (m) and the y-intercept (c). Here, the slope, m = 3 and the y-intercept, c = -6.

Draw the line

Now it's time to sketch the line. On the y-axis, mark the y-intercept, c = -6. From this point, use the slope to find other points on the line. The slope, m = 3, means move up 3 units and then move 1 unit to the right. Continue this to plot more points that lie on the line.

Shade the correct region

The inequality is \(y \leq 3x - 6\), this tells that the solution will be any y-value that is less than or equal to \(3x - 6\). So, the region below the line should be shaded to demonstrate the solution to the inequality.