Question

Sketch the graph of the inequality in a coordinate plane. $$ 3 x-y<0 $$

Short answer

The graph of \(3x - y < 0\) is a dashed line passing through the origin, with a slope of 3. The region above this line is shaded to show the solution to the inequality.

Step-by-step solution

Rewrite the inequality as an equality

Rewriting the inequality \(3x - y < 0\) as an equation will help us identify where the boundary line is. So, \(3x - y = 0\) or \(y = 3x\), a straight line passing through the origin is formed.

Identify a test point

We have to select a test point that does not lie on the boundary line. The origin is generally a good test point, but since it's on the line here, we'll use another point. Let's take (0,1) as the test point.

Substitute Test Point Into the Inequality

Placing the point (0,1) into the inequality gives \(3*0 - 1 < 0\), or -1 < 0, which is true. Thus, the region containing the point (0,1) is the solution to the inequality.

Sketch the graph

The graph would be a straight line passing through the origin, with a slope of 3. Also, as the inequality is less than 0 and not less than or equal to 0, the line should be dashed to indicate that the points exactly on it are not part of the solution. Since our test point (0,1) which lies above this line fulfills the inequality, we can shade the region above the line to show the solution to the inequality.