Question

Sketch the graph of the inequality. $$y-x \leq 11$$

Short answer

The solution is the line \(y = x + 11\) and the area below this line.

Step-by-step solution

Write the inequality in slope-intercept form

Initially the inequality is in the standard form. By rearranging it, we can write it in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In our case, it transforms into \(y \leq x + 11\). Here, 1 is the slope and 11 is the y intercept.

Plot the line \(y = x + 11\)

Start by plotting the point at the y-intercept (0,11). Then, use the slope to find another point on this line. Slope is rise over run, so the slope 1 implies moving 1 unit up and 1 unit to the right from point (0,11) to plot the second point (1,12). Repeat this as much to get enough points for the line, and then draw the line through these points.

Shade the region represented by the inequality

Because our inequality is \(y \leq x + 11\), this means y values are less than or equal to line \(y = x + 11\). Therefore, shade all the area below the line to represent this.