Question

Sketch the graph of the inequality. $$x+6 y \leq 12$$

Short answer

The graph of the inequality \(x + 6y \leq 12\) is a line drawn through the point (0,2) with a negative slope of \frac{1}{6}. The region that satisfies the inequality (i.e., below the line or on the line) is shaded.

Step-by-step solution

Write the inequality as an equation

In order to graph the inequality \(x + 6y \leq 12\), it is beneficial to first graph the linear equation \(x + 6y = 12\). This equation is in the standard form of a linear equation, \(Ax + By = C\). Rewrite the equation in the slope-intercept form, \(y = mx + b\), to easily identify the slope and y-intercept. Doing this gives \(y = -\frac{1}{6}x + 2\). Thus, the slope of the line is -\frac{1}{6} and the y-intercept is 2.

Plot the line

Draw the y-intercept on the graph at point (0,2). Since the slope is -\frac{1}{6}, for each unit increase in x, y will decrease by \frac{1}{6}. This calculated slope will help in drawing the line accurately on the graph.

Shade the region

The graph of the line \(x + 6y = 12\) represents all points which satisfy the equation exactly. However, the original inequality was \(x + 6y \leq 12\). This is satisfied for all points either on the line or below it. Thus, shade the region below the line to show this.