Question

Sketch the graph of the inequality. $$x-y \geq 4$$

Short answer

The graph of the given inequality is a line passing through points (0, -4) and (4, 0), and the area below this line (excluding the origin) is shaded to represent the solution set of the inequality.

Step-by-step solution

Convert inequality to equation

To sketch the graph, first convert the inequality into an equation. The equation corresponding to the inequality \(x - y \geq 4\) is \(x - y = 4\).

Plot the graph for the equation

The equation \(x - y = 4\) is in the standard form of a linear equation, so choose two arbitrary values for \(x\) and calculate corresponding \(y\) values. For instance, let's take \(x = 0\), then, \(y = -4\). Alternatively, if \(x = 4\), then \(y = 0\). Now, plot these points (0, -4) and (4, 0) to draw the line on the graph.

Shade the solution area

The inequality is greater than or equal to, therefore the line itself is included in the solution. Express the line with a solid line. To decide which side of the line to shade, choose a test point not on the line, for example, the origin (0,0). Substitute these values into the inequality. It gives \(0-0 \geq 4\) which is false. This implies the test point lies outside the solution set, thus shade the region which doesn't include the origin.