Question

Sketch the graph of the inequality. \(y-3 x \geq 2\)

Short answer

The solution to the inequality \(y-3x \geq 2\) can be visualized by sketching a solid line of y = 3x + 2 and shading the area above the line.

Step-by-step solution

Rewrite the Equation

The first step is to rewrite the inequality in slope-intercept form (y = mx + c) by isolating y. This gives us \(y \geq 3x + 2.\)

Determine the Slope and the Y-intercept

The slope (m) is the coefficient of x, which is 3, and the y-intercept (c) is the constant term, which is 2.

Graph the Line

Plot a line y = 3x + 2 on graph as a reference line. Starting from the y-intercept (0,2), for every 1 unit movement to the right on the x-axis, move 3 units upwards to locate another point on the line.

Shade the Region

As the inequality is 'greater than or equal to', the line y = 3x + 2 should be a solid line indicating that the points on the line are included in the solution set. Since y is 'greater than', we will shade the region above the line.

Verify Solution

Take a test point not on the line (like the origin(0,0)) in the shaded region. If the inequality is satisfied, the shading is correct. Substitute x = 0, y = 0 into the inequality, we get 0 ≥ 2 which is false, confirming that the region above the line is the solution set.