91Ó°ÊÓ

First, we need to rewrite the inequality in slope-intercept form ($$y = mx + b$$), where m is the slope and b is the y-intercept. In this case, we are given: $$4x - y + 12 > 0$$ Subtract $$4x$$ and add $$y$$ to both sides to get: $$y < -4x + 12$$

Now, let's identify the slope and y-intercept from the inequality $$y < -4x + 12$$. The slope ($$m$$) is the coefficient of $$x$$: $$m = -4$$. The y-intercept ($$b$$) is the constant term, which is $$12$$.

Now that we have the slope and y-intercept, we'll graph the boundary line, which is the line that represents the equation: $$y = -4x + 12$$ Plot the y-intercept (0, 12) and use the slope to find another point on the line. Since the slope is -4, from the y-intercept we can go down 4 units and to the right 1 unit (or up 4 units and to the left 1 unit) to find another point on the line. Connect these points with a straight dashed line, as the inequality is 'greater than' and not 'greater than or equal to'. Now choose a test point to see which side of the line should be shaded. A common test point is (0, 0), the origin, unless the boundary line intersects the origin. In that case, choose another simple point, like (1, 0) or (0, 1). For our inequality, the origin is not on the boundary line, so we can use (0, 0) as our test point. Plug the coordinates into the inequality: $$0 < -4(0) + 12$$ $$0 < 12$$ Since the inequality is true, the region that contains the test point should be shaded.

As we determined in step 3, the region that contains the test point (0, 0) should be shaded. Since (0, 0) is below the boundary line, shade the region below the dashed line to represent the solution to the inequality. And that's it! The graph of the inequality $$4x - y + 12 > 0$$ is complete.